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TRANSACTIONS OF THEAMERICAN MATHEMATICAL SOCIETYVolume 281. Number 2, February 1984

BINOMIAL COEFFICIENTS AND JACOBI SUMS

BY

RICHARD H. HUDSON AND KENNETH S. WILLIAMS1

Abstract. Throughout this paper e denotes an integer > 3 and p a prime = I

(mod e). With/ defined by p = ef + 1 and for integers r and s satisfying 1 *s s < r

=c e — 1, certain binomial coefficients (r¡¡) have been determined in terms of the

parameters in various binary and quaternary quadratic forms by, for example, Gauss

[13], Jacobi [19,20], Stern [37-40], Lehmer [23] and Whiteman [42,45,46].In §2 we determine for each e the exact number of binomial coefficients (JÍ) not

trivially congruent to one another by elementary properties of number theory and

call these representative binomial coefficients. A representative binomial coefficient

is said to be of order e if and only if (r, j) = 1.

In §§3-4, we show how the Davenport-Hasse relation [7], in a form given by

Yamamoto [SO], leads to determinations of n(p~*)/m in terms of binomial coeffi-

cients modulo p = ef + 1 = mnf + 1. These results are of some interest in them-

selves and are used extensively in later sections of the paper.

Making use of Theorem 5.1 relating Jacobi sums and binomial coefficients, which

was first obtained in a slightly different form by Whiteman [45], we systematically

investigate in §§6-21 all representative binomial coefficients of orders e =

3,4,6,7,8,9,11, 12, 14, 15, 16, 20 and 24, which we are able to determine explicitlyin terms of the parameters in well-known binary quadratic forms, and all representa-

tive binomial coefficients of orders e = 5,10, 13, 15, 16 and 20, which we are able to

explicitly determine in terms of quaternary quadratic decompositions of 16/7 given

by Dickson [9], Zee [51] and Guidici, Muskat and Robinson [14]. Some of these

results have been obtained by previous authors and many new ones are included.

For e = 1 and 14 we are unable to explicitly determine representative binomial

coefficients in terms of the six variable quadratic decomposition of 72 p given by

Dickson [9] for reasons given in §10, but we are able to express these binomial

coefficients in terms of the parameter x, in this system in analogy to a recent result

of Rajwade [34],

Finally, although a relatively rare occurrence for small e, it is possible for

representative binomial coefficients of order e to be congruent to one another

(mod p). Representative binomial coefficients which are congruent to ± 1 times at

least one other representative for all p = ef + 1 are called Cauchy-Whiteman type

binomial coefficients for reasons given in [17] and §21. All congruences between

such binomial coefficients are carefully examined and proved (with the sign ambigu-

ity removed in each case) for all values of e considered. When e = 24 there are 48

representative binomial coefficients, including those of lower order, and it is shown

in §21 that an astonishing 43 of these are Cauchy-Whiteman type binomial coeffi-

cients. It is of particular interest that the sign ambiguity in many of these con-

gruences does not arise from any expression of the form n(p~1)/m in contrast to the

case for all e < 24.

Received by the editors February 8, 1983.

1980 Mathematics Subject Classification. Primary 10-02, 10C05, 10B35; Secondary 12C25.Key words and phrases. Binomial coefficients (mod p) in terms of binary and quaternary quadratic

forms, Jacobi sums.

'Research suported by Natural Sciences and Engineering Research Council Canada Grant No. A-7233.

©1984 American Mathematical Society

0002-9947/84 $1.00 + $.25 per page

431

License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use

432 R H. HUDSON AND K. S. WILLIAMS

1. Introduction and summary. Throughout this paper e denotes an integer > 3 and

p a prime = 1 (mod e). The integer/ is defined by p = ef + 1. For integers r and í

satifying 1 < s < r < e — 1, certain binomial coefficients

(1.1) \sf

have been determined modulo p by, for example, Gauss [13], Jacobi [20], Stern

[37-40], Lehmer [23], and Whiteman [42,45,46] in terms of representations of p by

certain quadratic forms. The first result of this kind is due to Gauss [13, Vol. 2, p. 90]

who showed that for e - 4, p = 4/ + 1 = a2 + b2, a = 1 (mod4),

(1.2) (y) = 2a (modp).

Emma Lehmer [23] used Jacobsthal sums to obtain congruences for (2j) and (3{)

when p = 5f + 1 in terms of the system I6p = x2 + 50m2 + 50u2 + 125w2, xw =

v2 — Auv — u2, x = 1 (mod 5), introduced by Dickson [8].

In this paper we systematically use Jacobi sums to obtain congruences modulo

primes p = ef + 1 for binomial coefficients of the type (1.1). These include old as

well as new ones. The cases treated in §§6-19 are e = 3,4,5,..., 16. In §§20-21 we

handle in some detail the cases e — 20 and e — 24, relying heavily on recent

evaluations by, e.g., Berndt and Evans [4] of bidecic and biduodecic Jacobi sums.

Our results are obtained in terms of the parameters in the following Diophantine

systems:

(1) p = a2 + b2, a = l(mod4), 4|<?,

(2) p = x2 + 3y2, jt = l(mod3), 3|<?,

(3) 4p=A2 + 2W2, ,4 = 1 (mod 3), 3|e,

(4) 16/7 = x2 + 50u2 + 50t)2 + 125w2, xw = v2 - 4uv - u2,

x = 1 (mod 5), 5\e,

(5) p = x2 + 7v2, x = 1 (mod 7), e = l, 14,

(6) lip = 2x2 + 42(x¡ + xj + x¡) + 343(x52 + 3x¡), jc, = 1 (mod 7),

e = 7,14,

(7) p = c2 + 2d2, c = l(mod4), e = 8,16,24,

(8) \6p = x2 + 26w2 + 26ü2 + 13w2, xw = 3v2 - Auv - 3u2,

jc = 9(modl3), e= 13,

(9) p = g2+\5h2, g=l(mod3), e=\5,

(10) p = x2 + 2u2 + 2v2 + 2w2, 2xv = u2-2uw-w2, x = 1 (mod 8),

u = v = w = 0 (mod 2), e = 16,

(11) p = e2 + 5f2, É> = a(mod5)if5|¿> and e =\b\ (mod5) if 5|a,

e = 20,


BINOMIAL COEFFICIENTS AND JACOBI SUMS 433

(12) p = u2 + 6v2, u = -1 (mod 4) if 3\b and u ee 1 (mod4) if 3|a,

24.

Although the sign of b is not fixed in (1) it is clear that for/) = 4/ + 1 there exists

at least one primitive root g(p) such that ge,/A = a/\b\ (mod p). For e — 20 and 24

some of our determinations require fixing g in this manner (see, in connection, [19]).

The following congruences are typical of those proved in §§6-21.

/=-A (modp = 3f+\)

y ee2û (mod p = 4/4- 1)

2/

/

U w(x2- I25w2)

2 \ ' 4(xw + 5uv)

(Jacobi [20]),

(Gauss [13]),

(mod p = 5/+ 1) (Lehmer[23]),

*\ =2x (modp = 6f+ 1),

2x (mod p = 7/+ 1) (Jacobi [19]),

(2/My)+($~, <-,=*♦.).(2/)=(-l)A/42c (mod/, = 8/+l),

(modp= 10/4-1),-x —w(x2- \25w2)

4(xw + 5uv)

II = -A (mod/>= 12/4- 1),

4/\l_/_ 3w(x2- 13w2)

// ~2\X 8(wx+ \3uv)

V,

f)=2a (mod 12/4- 1),

a = 1 (mod4) <=» 3 |6,

(modp= 13/4- 1),

AB = 0(mod5),'2g(modp=\5f+ 1),

2AA-9B8 (modP= 15/+1), ,4 =5 or-25 (mod 5)

2,4g - 18flg

,4+95(mod /7= 15/4- 1), A = -5 or 25 (mod 5),

k{. EE(-l)/2cor(-l)/+12c (mod/> = 16/+ 1)

according as b = 0 or 8 (mod 16),

(yH-4.-f'rff ) (N-/-IV+D.\ / / \ mz - vvz + 2«w /


434 R. H. HUDSON AND K. S. WILLIAMS

7)+ (*') =4(-l)7x (modp=l6f+\),

1), 5\bor5\a (Whiteman [45]),

(modp = 20f+ 1)

for the solution (x, -v, u, -w) of (4),

= (_!)/+*/«+1¿ (modp = 24/+l),

ee(-1)/2m (mod/? = 24/+ 1).

(5/)

mIn so far as possible we determine all binomial coefficients for the cases consid-

ered if they can be given in terms of the systems (1)-(12). Binomial coefficients

which are not treated are related to the parameters in more complicated quadratic

partitions. In some cases, see §§18-21, we are able to determine these binomial

coefficients up to sign in terms of the parameters in (1)-(12).

For each ei there are $e — $(e — 2) binomial coefficients of the type (1.1) which

are of order e. In §1 it is shown by an application of a simple generalization of

Wilson's theorem that it suffices to determine N(e) of these binomial coefficients

(for large e, N(e) s e2/12) where N(e) is given explicitly by

, , , , i(e2-a)/12 if e = a (mod6), a = -3,0,1,4,(1.3) Afe = \ , ) :

[(e2 + a)/12 ifeEE«(mod6), a = -4,-1.

These N(e) binomial coefficients will be called representatives.

When e is composite, say e = mn, it is useful to have a congruence of the type

i> / o, f\a-,f\ ■ ■ ■ a,f$\ 4) n(p-$/"> = _Li—11-'I— (mod p)y ' bj\b2f\---b,f\ Vlw"P>

where the a¡, b¿ are integers between 1 and e — 1 inclusive. Such a congruence

follows from the Davenport-Hasse relation for Gauss sums [7] and a congruence of

Yamamoto [50]; see (3.11). For values of m and n for which m or n is small we show

in §4 that the expression on the right-hand side of (3.11) can be given in terms of

binomial coefficients (mod p). Together with known results for n(p~^/m (mod p) in

terms of representations of p by quadratic forms we deduce congruences (mod p )

relating certain binomial coefficients which are used in later sections. For example, if

p = 2mf + 1, it is shown in Theorem 4.1 that we have

(,,, 2(,-^E(_1),(<» + 2)/)/(<».2y>/) (modp).

Putting (1.5) together with a result of Emma Lehmer [23], see (4.5), we obtain

Corollary 4.1.1 which is used in §§9, 15, and 21. The results in §4 appear to us of



some interest, totally apart from their use in later sections. For example, an easy

application of (2.1), (2.2) and (3.11) gives

*,-.>/« =(_!/( »*/)/ ( !3y) (mod p = 48/+ 1).

We note that criteria for 3 to be a 16th power in terms of the parameters in

Diophantine systems is an open problem.

In 1840, Cauchy [5, p. 37] show that

o,, (THv) <—*=*/+!).and in 1965 Whiteman resolved the sign ambiguity in this congruence. Representa-

tive binomial coefficients which are congruent to ± 1 times another representative

modulo p for all p — ef + 1 are said to be of Cauchy-Whiteman type for reasons

given in [17] and §21. We systematically investigate all such congruences. The 27

congruences of this type given in Theorems 21.1 and 21.2 far exceed the number of

such congruences for all e < 24. Moreover, the congruences in Theorem 21.2 do not

arise (as do all other known Cauchy-Whiteman type congruences) as expressions of

the form («<'" 1)/m)',/> = mnf+ 1.

For/7 = 11/ + 1, 4/7 = a2 + 1 lb2, a ee 2 (mod 11), Jacobi [19] showed that

(M/(îfl-. ™For larger values of e with many representative binomial coefficients it is not

appropriate to list all such congruences, and we cite only one.

For/7 = 20/+ 1 =e2 + 5/2 (e = a (mod5) if 5 \b and e =\b| (mod5) if 5 |a) we

have

(fl(y)/(*).(-,r^<M,i,The sign may be given unambiguously in this congruence (and in many other

congruences in §20) only because of an important sign ambiguity resolution ob-

tained by Muskat and Whiteman [31] in determining the cyclotomic numbers of

order 20.

In all that follows we are heavily indebted to Berndt, Evans, Muskat, and

Whiteman for their pioneering work on Jacobi sums of higher orders.

2. The number of distinct binomial coefficients of the type (1.1). If m and n are

positive integers such that m + n = e, then a simple modification of Wilson's

theorem yields

(2.1) m/!«/! = (-ir/_,=(-l)n/"1 (mod/7).



Making use of (2.1) and the elementary property (£) = (a-h) of binomial coeffi-

cients, we deduce that for 1 < s < r < e — 1 we have

« (i)-((r-^)-<-H{:-ä(-,r*((-ä

s<-i»"((v")//)s(-i)"((e":/+s,/)<mo<""-

If r 7e 2s, 2r — s ¥= e, r + s ¥= e then the entry pairs in the six coefficients in (2.2)

are distinct and we call (2.2) a 6-cycle. If exactly one of r = 2s, 2r — s = e,

r + s = e holds, then there are three distinct entry pairs in the coefficients in (2.2)

and we call (2.2) a 3-cycle. Finally, if at least two of r — 2s, 2r — s = e, r + s = e

hold, then in fact all three hold so that e = 3s and the coefficients in (2.2) reduce to

the 1 -cycle (2//).

The number TV, of 1-cycles is clearly 1 if e = 0 (mod 3) and 0 if e z 0 (mod 3). The

number N3 of 3-cycles is the number of pairs (r, s) satisfying exactly one of r = 2s,

2r — s = e, r + s = e. Thus 7v"3 is the number of integers t satisfying 1 < t < e/2,

t i= e/3, so that

e/2-2 ifeEEO (mod6),

(e- l)/2 if e = 1 (mod6),

e/2 - 1 if<?EE2 (mod6),

(2'3) ^3" I (c - 3)/2 ife = 3 (mod6),

e/2- 1 ifi>EE4 (mod 6),

(e- l)/2 if£>EE5 (mod6).

The number N6 of 6-cycles is now easily deduced from the values for TV, and N3 and

the identity

(2.4) TV, + 3/V3 + 6/V6 = {(e - \)(e - 2).

Since the coefficients in (2.2) are congruent (mod/7) it suffices, for each e, to

determine N(e) = TV, + 7V3 + N6 of them. For the convenience of the reader Table 1

is given summarizing the above and indicating the representative to be chosen from

each cycle.

A representative binomial coefficient of the type (1.1) with (r, s) > 1 is the same

as the lower order binomial coefficient (rs\j¡), where

rx = r/ (r,s,e), s, = s/ (r, s,e), ex = e/(r, s, e) < e,

/, = (r, s,e)f, p = exfx + 1. Henceforth, a representative binomial coefficient will be

said to be of order e only if it is not the same as one of lower order. It is easy to see

that if 5(e) denotes the number of representatives with lower order ones excluded,

we have

5(3) = 5(4) =1, 5(5) = 5(6) = 2, 5(7) = 5(8) = 4,

(2.5) 5(9) = 5(10) = 6, 5(11) = 10, 5(12) = 8, 5(13) = 14,

5(14) =12, 5(15) = 5(16) = 16, 5(20) = 24, 5(24) = 33.


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We close this section with a lemma which will be useful when 5(e) > 1 in that it

makes it possible to further reduce the number of binomial coefficients which must

be treated separately.

Lemma 2.1. Ifg, h, k are integers satisfying 1 < A < g < e — 1, \<h<kê— 1,

e — k < g — h, then

gf\l(e~ g)f\ _, i«g+k)f(kf\¡(e-k)f

3. Basic properties of Gauss and Jacobi sums. For a positive integer n we set

f„ = exp(27T/'//i). For (a, e) = 1, we define the automorphism oa by

(3.1) Vf.-C, °a-P-»Pa,

where 5 denotes any of the <i>(e) prime ideals dividing p in (?(fe). Let g be a

primitive root (mod p) such that g(p~i)/e = ff (mod 5). We define a character xe

(mod />) of order e by

|C iîxÔ(modp),x^-ê = ^(modP),

{ } XÁX) [0 ifxEE0(mod/7),

so that xe(a>) = ¿V If * z 0 (mod p), the index of x with respect to g, written

indg(;c), is the unique integer b such that x = gh (mod /7), 0 < b </7 — 2. Clearly

Xe(*) — fèndg(x)- Let r and s denote positive integers. The Gauss sum Ge(r) of order

e is defined by

p-\ p-\

Ge(r) = s xWp = 2 $rd°(x-x = 0 x=\

The Jacobi sum Je{r, s) of order e is defined by

p-\ p-\

Je(r>s)= 2 Xe(x)x'eO - X) = 2 t**W*W-*\x = 0 x=2

Gauss and Jacobi sums are related by

(3-3) Je(r, s) = Ge(r)Ge(s)/Ge(r + s),

provided e does not divide r, s, or r + s. Moreover (see, for example, Muskat [30]),

we have

(3.4) Je(r, s) = Je(s, r) = (-l)"/,(-* - s, s),

(3.5) 7e(r, s)/e(-r,-s) =p (r, s, r + s s 0 (mod e)),

(3.6) G,(r)Ge(-r) = (-l)*/» (r Z 0 (mod e)),

(3.7) Jm(r, s) =/e(r/î, s«) ûe = mn,

and if e is prime and e J r, s, or /• + s, then

(3.8) Je(r,s) = -\ (mod(l-U2).



An important relation involving Gauss sums is provided by the Davenport-Hasse

relation [7], namely,

• .,, , Ge(tn)l\"z\Ge(mj)(3.9) Cd'(,"= ' i , \ . i=l,2,...,m-l.

Further, it follows from the work of Yamamoto [50] that if IIy=¡{Ge(/)}'> (c an

integer) is a unit of Q(Çep ), then

e-I

and

(3.10) nW^r^ïo/^ (modp).j=\ 7=1

Applying this to (3.9) we obtain

ntf\l\"Z\(mjf)\(3.11) „(/>-!>'/«= /-lV y (mod/7).

n^o((m/ + 0/)!

Finally, it follows from Stickelberger [41] that if et r, s or r + s, and (Je(r, s))

denotes the ideal generated by Je(r, s), then

(3.12) <^.*))= n p,t=\

U,e)=\

{rr'/e) + {st-'/e}<\

where t'x denotes the inverse of t modulo e, and { } denotes, as is customary, the

fractional part of the quantity inside the braces.

4. n(p~1)/m as a product of binomial coefficients (mod p). In this section for values

of m and n for which either m or « is small and t = 1 we show that the expression on

the right-hand side of (3.11) can be expressed as a quotient of products of binomial

coefficients (mod/7). Making use of known results in the literature for nip~])/m

(mod p) in terms of representations of p by quadratic forms, we deduce congruences

(mod p) relating certain binomial coefficients of the type (1.1). These congruences

will be used in later sections.

Throughout this section we have p = mnf + 1. Taking n = 2 in (3.11) and

appealing to (2.2), we obtain for each t = 1,2,... ,m — 1 and/7 = 2mf + 1,

2<-»^E=(-ir(^)/(^) (mod/7).

Binomial coefficients (mod p = e/+ 1), when e is composite, are often related to

one another by powers of n(p~1)r/m where « is a divisor of e (not necessarily prime).

For e < 12 we are able to determine these interrelationships by simply taking t = 1

in (3.11) so we may appeal directly to results in this section. Beginning with e = 14

in §17 the number of powers of n(p~x^'/m which need be considered becomes



sufficiently large that it is convenient to use the full strength of the congruence

(3.11). We postpone this generalization for now and it will be understood throughout

this section that t = 1 in applying (3.11).

Taking n = 2 in (3.11), and appealing to (2.2), we obtain

Theorem 4.1. If p = 2mf + 1 is prime then

(4.1) 2<>->/»S(l)'( {m~l)f)/[r¡ff) (mod,).

When m — 2 we have, as is well known,

(4.2) 2ip-l)/2=(-l)f (mod/7 = 4/+ 1).

When m = 3, Theorem 4.1 gives

(4.3) (-!)'( 2/)=2<'-'>/3(y) (mod/7 = 6/+l).

As p is a prime = 1 (mod 3), there are integers x, y such that

(4.4) p = x2 + 3y2, x = \ (mod 3).

The determination of 2(p~ l)/3(mod p) in terms of x and v is given by

(4.5) 2("-'>/3 =

1 (mod p) ifv = 0(mod3)

x + 3v

x

x

3v

3jx + 3 v

(mod/?) if v = 1 (mod 3)

(mod/7) ifv' EE 2 (mod3)

(Jacobi [19]),

(Lehmer[23]).

Primes p = 1 (mod 3) are also expressible in the form 4p = A2 + 2752, A = 1

(mod 3), where A and 5 are related to x and v in (4.4) by

(4.6)

A = -2x, B=±2y if v ee 0 (mod 3),

y4=x + 3y, 5=±3L(*-v) if y ee 1 (mod 3),

y4 = x - 3v, 5 \(x + y) ifv ee 2 (mod 3).

Thus (4.5) may be reformulated in terms of A and 5 as follows:

l(mod/7) if ,4 =5 = 0(mod2),

y4 + 95(mod/7) if,4 =£ee 1 (mod2),/< ee 5 (mod4),

A — 95——— (mod/7) if ^ ee 5 ee 1 (mod2),A = -5 (mod4).A + yß

(4.7) 2('-')/3ee.

Combining (4.3) and (4.5) we obtain



Corollary 4.1.1. Ifp = 6/+ 1 = x2 + 3v2, x = 1 (mod 3), is prime, then

(-l/jyjOnodp) */v = 0(mod3),

(-l/f^fyjimod/,) //>>EEl(mod3),

(-1)7^ (2/) (m°d p) i/vEE2(mod3).

3/\_/


Corollary 4.1.2. Ifp = 6/+ 1, with 4p = A2 + 2752, A = 1 (mod3), is prime,

then

>

(-!)'( y) (mod/7), if A =5 = 0 (mod 2),

(-^J^rff (2/) (mod^)' '/^=5EEl(mod2),>i=5(mod4),

(_1)/l^lf ( /] (mod ^ lfA=B=l (mod2)> ^ = "* (mod4)-

(4.8)

When m — 4, Theorem 4.1 gives

(yj^-l)'^)/4^) (mod/7 = 8/+l).

As p = 1 (mod 8) there are integers a and b such that

(4.9) p = a2 + b2, a = l(mod4), ¿7 = 0(mod4),

and by a result of Gauss [13, Vol. 2, p. 89] we have

2O--0/4 =(_!)*/< (mod/7).

Hence from (4.9) and Lemma 2.1, we have

Corollary 4.1.3. If p = Sf + I = a2 + b2 (a = I (mod4), ¿7 = 0 (mod4)) is

prime, then

(*)-<-.)-$) (nMd,,


(4.10) (y) ^i-l/^-)/5!^) (mod/7 = 10/+ 1).

As p = 1 (mod 5) there are integers x, u, v, w such that

_2

(4.11)I6/7 = x2 + 50m2 + 50i72 + 125w2, x = 1 (mod 5),

xw = v2 — 4uv — u2.



It is known (see, for example, [49, p. 544]) that (4.11) specifies x uniquely. If x = 0

(mod2), Lehmer [21] has shown that 2(p~X)/i = 1 (mod p). Moreover, if x = 1

(mod 2), then there is a unique solution (x, u, v, w) of (4.11) satisfying

(4.12) meeO (mod2), x + u - v = 0 (mod4),

and for this solution we have 2(p~1)/5 = a(x, u, v, w)(mod p) where

„ ,„, , v w(l25w2 - x2) + 2(xw + 5mu)(25w - x + 20m - 10«)(4.13) a(x, u,v,w) = —-zr----■

w(l25w2 - x2) + 2(xw + 5mu)(25w - x - 20m + 10»)

Combining these we have

Corollary 4.1.4. If p = 10/ + 1 is prime, and (x, u, v, w) is a solution of (4.11)

satisfying (4.12) we have

4f\-f

Ï 2/1 if x = 0 (mod 2),

(-l)fa(x,u,v,w) {. (mod/7) if x = 1 (mod2),

w/iere a(x, u, v, w) is given by (4.13).


(4.14) (5/)s(-l/2f-^(^) (mod/7=12/+l).

Since 2(i"l)/2 ee (-l)/(mod /,), we have

(4.15) (/H2<'~1>/3)2(2/) (m0d^-

Appealing to (4.5), we obtain

Corollary 4.1.5. Ifp = 12/ + 1 = x2 + 3 v2 (x = 1 (mod 3)) is prime, then

6/1

5/\_/

. (mod/7) z/y = 0(mod3),

^f(26^)(mod/7) ,/^l(mod3),

7^(2/) (mod^) ^ = 2(mod3).


(4.16) (y)s(-l/2<'-' 7/7/

2/(mod/7= 14/+ 1).

The determination of 2(p"1)/7 (mod /7) has been given by Nashier and Rajwade [33].

Since this determination is extremely complicated, we just illustrate it below for the

case when 2 is a seventh power (mod p).



Corollary 4.1.6. Let p — 14/ + 1 be a prime. Then there are integers x,,... ,x6

such that

(4.17)

72^ = 2x2 + 42(x2 + xj + x2) + 343(x5 + 3x6)2,

\2x\ - 12x2 + \41x\ - 441x62 + 56x,x6

+ 24x2x3 — 24x2x4 + 48x3.X4 + 98x5x6 = 0,

12x32 - 12x2 + 49x52 - 147x62 + 28x,x5 + 28x,x6

+48x2x3 + 24x2x4 + 24x3x4 + 490x5x6 = 0,

with x, EE 1 (mod7). All the solutions of (4.17), except the two trivial solutions

(x,, x2, x3, x4, x5, x6) = (-6/, ±2m, ±2m, =í=2m,0,0), where p — t2 + 7m2, t = 1

(mod7), have the same value of x,. // x, ee 0 (mod 2), then 2 is a seventh power

(mod p), and we have

(4.18) (y)-(-l/|V) (mod/7).

Example. We illustrate Corollary 4.1.6 by taking p — 673 so that /= 48 and

x, = 22 = 0 (mod 2) (see [47, p. 1136]). In agreement with (4.16), we have

f= (24888)ee346 (mod 673),

(-1)/(27/) = (3966)"346 <m°d673)-


7/(4.19)

From Lehmer [23, p. 66] we have

■ /(-l)/2('-,,/'(^) (mod/,= 16/+l).

(4.20) 2<i--')/8 = J

+ 1 ifZ7 = 0(modl6),

+ b/a if b = 4 (mod 16),

-1 if b = 8 (mod 16),

-b/a \ib= 12 (mod 16),

where a and b are defined as in (4.9). Combining (4.19) and (4.20) we obtain

Corollary 4.1.7. Let p = 16/+ 1 = a2 + b2 (a = 1 (mod4), 6 = 0 (mod4)) ¿>e

prime. Then

7/\_

/ -(-•/(ira (-'>■Since expressions for 2</,_1)/m (mod /?) are also known for m = 10, 12, 15, 16, 20,

24, 32 and 40 (see [23, p. 70; 18]), similar congruences to those given in Corollaries

4.1.1-4.1.7 can be deduced.



Next we take n = 3 in (3.11) to obtain

Theorem 4.2. Ifp = 3m/+ 1 is prime then

When m — 3, Theorem 4.2 gives

(4.21) (y)EE3<"-"/3(2^) (mod/7 = 9/+l).

By a result of Lehmer [23, p. 67] (see also [48, p. 279]) we have

l(mod/7) if5 = 0(mod3),

(4.22) 3<*-0/3A — 95

(mod/7) if5EE l (mod3),,4 + 95

A + 95

,4 -95

^ + 95ir^(mod/7) if5 = 2(mod3),

where

(4.23) 4/7 = A2 + 2752, ^ = 1 (mod 3).


Corollary 4.2.1. Ifp = 9/+ 1 is prime then

( 2/) - ( 5f) (mod ̂ ifB = ° (mod 3)'

= ^H(5/)(mod/7) ,/5ee l(mod3),

^7rlf(5/)(mod/,) '/5 = 2(mod3)-

When w = 4, Theorem 4.2 gives

(4.24) (6/)-(-3)('",)/4(y) (mod/,= 12/+l).

From the work of Gösset [15] we have

(p-\)/a _ [ 1 (mod p) if ¿) = 0 (mod3),

-l(mod/7) ifa = 0(mod3),

where a = 1 (mod4), ¿7 = 0 (mod2) ((-3)^"1)/4 = ± 1 (mod p) as/7 = 1 (mod 12)).

Combining (4.24) and (4.25) we have

Corollary 4.2.2. If p = I2f+ \ = a2 + b2 (a = I (mod4), ¿7 = 0 (mod2)) is

/7n'me then

V\ -/

f^j (mod p) i/¿7 = 0(mod3),

M (mod p) if a = 0 (mod 3).




(4-26) (/)S3('"I)/5(3/) (mod^15/+1)-

An explicit determination of 3(p~1)/5 has been given in [49, Theorem 2]. Using this

together with Jacobi sums of order 15 given by Muskat [30] an explicit determina-

tion of (J) and (Yf) is obtained in §18.

Taking n = 4 in (3.11), and appealing to (2.2), we obtain


4(,-„/,-/2/\/3/W4/xxvirrni y) ™/Taking m = 3 in Theorem 4.3 we obtain


(6/)=2<-'>/3(y) {moápy

We note that it is possible to obtain a simpler form for the determination of

4<p i)/m (m0(j pj by using (2.1) together with (3.11). In particular, we have

^(p-\)/m =4/!(-l)m/ '(2m/)!

uu ^,wm /!((«+l)/)!((2«+l)/)!((3«+l)/)!II {(mj+ 1)/)!

7 = 0

/(m + 3)/W(m + 2)/

(-l)/4/!(2m/)!(m-l)/! _, „/ \ f }\ f:(-!)'

/!((m+l)/)!((2m+l)/)! / (m + 3)/W (2m + 1)/|

Thus we have obtained the following variation of Theorem 4.3.


4P-Wm^{_x)fUrn + 3)f\i(m + 2)f\^ I (m + 3)/W(2m+l)/\

(mod p).

Although Theorems 4.1 and 4.4 clearly give the same result for m = 3, this is not

the case in general.

The following congruence, which will be referred to again in §14, is particularly

interesting, since it shows that representative binomial coefficients may be identical

modulo p = e/+ 1. Several of these identities are established in §21. It would be

interesting to know for which values of e such congruences are possible.



(mod p ).

/(SMS)/(?) <-'>■

(mod p )

Corollary 4.4.1. Ifp — 12/+ 1 is prime then

(5/h:í)Proof. Using (3.4) together with Theorems 4.1 or 4.4 (appealing to (2.2) to see

that (-\)f(6/) = C/) (mod p) in the latter case),

(î)/(ï)-(ï)/(rOn the other hand,

is an immediate consequence of Lemma 2.1 with g = 4, h = 2, k — 6.

We remark that one can also obtain the last step in the proof by expanding in

terms of factorials using (2.1). This corollary is important to us, as using it, other

representative binomial coefficients of order 12 are determined in §14 using the

simple determination of (2/),p = 3/ + 1, given by, e.g., Jacobi [20]. Taking n = 5 in

(3.11), and appealing to (2.2), we obtain


(m + 2)/\ / (3m - l)/\ / / (m + 4)/\ / (3m + 1)/)

/ f\(m + 2)f}' \

Taking m = 3 in Theorem 4.4 we obtain

5(/>-l)/m =

5/ (m + 4)/(mod p ).

(4.27) 5-M = s^-'/

= 5< \J^ (mod/7=15/+l)

By a theorem of Williams [48, pp. 282-283] we have

(4-28)5</>-n/3

1 (mod p ) if^5EE0(mod5),

A + 95Q (mod/7) if A = B or -25 (mod 5),95

95(mod/7) if A = -5 or 25 (mod 5),

A + 95

where 4/7 = A2 + 2752, A = 1 (mod 3). Combining (4.27) and (4.28) we obtain


If"(mod/7), if AB = 0 (mod 5),

^H 72f\ (mod p), if A =Bor -25 (mod 5),

(mod p), if A EE -5 or 25 (mod 5).

2/

^4 + 95,

A-9B (If

A + 9B\2f



5<"~1)/4( 3°/) (mod/, = 2°/+l).

Finally, taking m = 4 in Theorem 4.4, we obtain

(7)Since

5(í-D/4 = f1 (mod/7), if¿>EE0(mod5),

|-l(mod/7), if a ee 0 (mod5),

where a = 1 (mod 4), we have

Corollary 4.5.2. Ifp = 20/ + 1 is prime then

io/\_/

^ (mod/?) ifb = 0(mod5),

(mod/7) if a = 0(mod5).3/

Corollary 4.5.2 was first proved by Whiteman [45], although the congruence

(l0/) ee ±(3°/) (mod p) had been established by Cauchy [5,p. 37] one hundred and

twenty-five years earlier.

5. The basic theorem. We prove the following theorem which shows how each

binomial coefficient of type (1.1) can be determined modulo P by means of Jacobi

sums. This theorem provides a basic tool which will be used through the rest of the

paper.

Theorem 5.1. If p = ef + 1 is prime and r, s are integers such that 1 < s < r < e —

1, then

(5.1) (^)=(-ir/+Ve(r,e-s) (mod5).

Proof. Since

Xe(x)=x' (mod 5)

we have

p-\

Je(r, e - s) = 2 *r/0 - *T~S)i (mod 5)x=\

p-\ (e-s)f1-1 ye-s)j /, . v

2*r/ 2 (-1)' (e_i)/x':=I f = 0 \ Í /

2Vi)'((e~5)/) 'ïx*+:=n \ / / r=i

However,

(5.2)_[0(mod/7) if Â: s 0 (mod p - 1),

x=i1 (mod p) if A: = 0 (mod p — 1),


448 R H HUDSON AND K. S. WILLIAMS

so that we have, appealing to (2.2),

(5.3) Je(r,e-s)JeÍ (-1)i = 0

l(e-s)f)

I (e - s)f\ _, ,vf+\[rf

We note that Whiteman [45] has already proved a result similar to, but not exactly

the same as Theorem 5.1. Letting ß = e2v,/e be replaced by g¡ for a primitive root g

of p = e/+ 1 our Je(r, s) becomes Whiteman's \pr s. In Lemma 6 of [45] Whiteman

showed that

(i) 4>rs = 0(modp) (r + s<e),

I (2e - r - s)f\(ii) «rV,*-- (e-r)f j ^mod p^ (r + s>e).

In view of (2.2), condition (ii) can be rewritten in the simpler form

,s/+l| rf(ii)' *(r.í)=(-l) """ Lj^J (mod/7).

In later sections we will refer again to Whiteman's very useful Lemma 6.

6. e = 3. There is a single representative binomial coefficient of order 3, namely

(2f). With A and 5 defined as in (4.23), we choose P — (it), where

(6.1) 'ÏÏ = \{A +35/^3),

so that P1/7. It is well known that

(6.2) y3(l,l) = 77,

(see, for example, [4, p. 357]), so by (6.1) and (6.2)

(6.3) /3(2,2) = v = ^(a - 35/^3) ee^I (mod 77).

Hence, by Theorem 5.1 (with e = 3, r = 2, s = 1) we have (as/is even),

(6.4) y ee -y3(2,2) ee -A (mod 77).

As (2J) and -A are both rational integers, and ir\p, we have

Theorem 6.1. If p = 3/+ 1 is prime and A is given uniquely by 4p = A2 + 2752,

/I ee 1 (mod 3), then

y) =-¿ (mod/7).

This result is due to Jacobi [20]; see also Whiteman [42] and von Schrutka [35].

Thus, appealing to (4.6), Theorem 6.1 can also be given in the form



Theorem 6.2. If p = 3/+ 1 is prime and x is given uniquely by p = x2 + 3v2,

x ee 1 (mod 3), then

2x (mod p), if y = 0 (mod 3),

-x — 3v (mod p), if y = 1 (mod 3),

-x + 3v (mod p), if y = 2 (mod 3).

2/

/

7. e = 4. There is a single representative binomial coefficient of order 4, namely/

/(2/).

With a = 1 (mod 4), b = 0 (mod 2), we choose 5 = (77), where 77 = a + ¿7/', so that

P1/7. Then it is known that./4(l,2) = (-l)/+l77 (see, for example, [4, p. 361]), so

/+>=-/• n/+'/-„_ «..-ï =/■ n/+'-4(2,3) = /4(1,2) = (-1)' v = (-ir'(a - bi) =(-\)^l2a (mod 77).

Thus, by Theorem 5.1, we have

|2/) EE(-l)/+ly4(2,3)EE2a (mod 77),

and hence

Theorem 7.1. If p = 4/+ 1 is prime and a is given uniquely by p = a2 + b2, a = \

(mod 4), then

- 2a (mod p).

This is the result of Gauss mentioned in §1; see also Whiteman [42, p. 95].

8. e = 5. There are two representative binomial coefficients of order 5, namely

(¥) and (V). For convenience we set ß = f5. It is known that the ring of integers 5

or Q(ß) is a unique factorization domain [27]. In 5, p factors into primes as

(8.1) p — 77,772773774,

where 77 is any prime factor of p in 5 and mi = 0,(77) (i = 1,2,3,4). We can set

(8.2) n = axß + a2ß2 + a3ß3 + a4ß4,

where a„ a2, a3, a4 are rational integers (see, for example, [49]). Clearly a, + a2 +

a3 + a4 2 0 (mod 5), as 1 — ß \ 5, 5 \p. Replacing 77 by its associate 077, where a is

the unit of 5 given by

(8.3)

+ 1 if a, + a2 + a3 + a4 = 1 (mod 5),

-(ß + ß4) if a, + a2 + a3 + a4 = 2(mod5),

+ (ß + ß4) if a, + a2 + a3 + a4 ee 3 (mod5),

-1 if a, + a2 + a3 + a4 ee 4 (mod 5),

we may suppose that 77 = 1 (mod(l — ß)).

Replacing the new value of 77 by its associate ß-<"i+2a2+3a3+4a4)77i we may SUpp0se

further that

(8.4) 77 = 1 (mod(l-ß)2).


450 R. H. HUDSON AND K S. WILLIAMS

By a theorem of Stickelberger, see (3.12), we have

(8.5) /S(1,1)=0 (mod77,773),

so

(8.6) 4(1,1) = «77,773,

where m G 5. From (3.5), (8.1) and (8.6) we have

MM77|772773774 = ( M77,773 )( M77,773 ) = J¡( 1, 1 ) J$ ( 1, 1 ) =/7 = 77,772773774 ,

SO

(8.7) MM=1,

showing that m is a unit of 5, that is (see, for example, [36]),

(8.8) M=±(ß + ß4)V (A: = 0,±l,±2,...;/ = 0,1,2,3,4).

Now (8.7) guarantees that k = 0 in (8.8) so

(8.9) u=±ß' (/ = 0,1,2,3,4).

By (3.8), (8.4), (8.6) and (8.9), we have

±ß'=u

= M77,773 (mod(l — ß)2)

= 4(1,1) (mod(l-ß)2)

EE-1 (mod(l-ß)2),

so in (8.9) the minus sign holds with / = 0, that is, u = -1, giving

(8.10) 4(1,1) = -77,773.

We set

(8.11) 4(1. 0 = c,ß + c2ß2 + c3ß3 + c4ß4.

As 4(1,1) ee -l(mod(l -ß)2), by (3.12), we have

Je, +c2 + c3 + c4ee -1 (mod5),

*■ " ' \cl + 2c2 + 3c3 + 4c4ee0 (mod5).

Next, since ß - ß2 - ß3 + ß4 = /5 , we have

(c, -c2-c3 + c4)^5=(Ci -c2-c3 + c4)(ß - ß2 - ß3 + ß4)

-(1+c,+ c2 + c3 + c4) (mod(l-ß)4)

ee 2((c, + c4)(ß + ß4) + (c2 + c3)(ß2 + ß3) + 2) (mod(l - ß)4)

ee 2(4(1,1)+ 4(1,1) +2) (mod(l-ß)4)

= 2(4(1,1)+ 1)(/5(1,1) + 1) (mod(l-ß)4)

eeO (mod(l-ß)4),



SO

(8.13) c, - c2 - c3 + c4eeO (mod5).

Congruences (8.12) and (8.13) enable us to define integers x, u, v, w by

x = - (c, + c2 + c3 + c4), 5m = c, + 2c2 — 2c3 — c4,

(8.14)5 c c2 + c3 — 2c4, 5w = c, — c2

Using (3.5), (8.11), (8.12) and (8.14), it is easy to check that (x, u, v, w) is a solution

of

(8.15)16/7 = x2 + 50«2 + 50u2 + 125w2, x = 1 (mod 5),

xw = v2 — 4uv — u2.

(8.16)

From (8.14), we obtain

4c, = -x + 2m + 4v + 5w, 4c2 — -x + 4u — 2v — 5w,

c3 — -x — 4m + 2v — 5w, 4c4 = -x — 2m — 4v + 5w,

and so (8.11) and (8.16) give

(8.17) 4(1,1) = ^(x + M(2ß + 4ß2 - 4ß3 - 2ß4)

+ u(4ß-2ß2 + 2ß3-4ß4) + 5w{5).

Next, from (8.17), we deduce that

(8.18) 4(1,1)+4(4,4) = ^(x + 5w/5).

Since

(8.19) 4(1,1) =0 (mod 77),

by (8.5), we deduce from (3.4), (8.18) and (8.19) that

(8.20) 4(2,4) = 4(4,4) =\(x + Sw{E) (mod 77).

Hence, by Theorem 5.1, we have

(8.21) y ) =-4(2,4) = -!(* +5W5") (modTr).

It now remains to determine ^5" (mod 77) in terms of x, u, v, w.

Since

(8.22)

ß + 2ß2 - 2ß3 - ß4 = \iJ50 + IO/5 ,

2ß - ß2 + ß3 - 2ß4 = -iY50 - IO/5 ,

we obtain from (8.17) and (8.19):

(8.23) x + iMi/50 + IQ/5 + ivJsO - IO/5 + 5w/5 ee 0 (mod 77)



Also from (8.5) we have

(8.24) 4(1,1)=0 (mod773).

Applying the automorphism a2 to (8.24), we obtain

(8.25) 4(2,2) eeO (mod 77,).

Hence from (8.17) and (8.22) we have

(8.26) x - iU]¡50 - IO/5 + /o^50 + IO/5 - 5w/5 ee 0 (mod 77).

Adding (8.23) and (8.26) we obtain

(8.27) 2x + /(« + o)^50 + IO1/5 - i(u - v)]J50 - 10/5 ee 0 (mod 77).

Taking the term 2x over to the right-hand side of (8.27) and squaring, we obtain

after some simplification,

(8.28) 10(m2- uv- v2)fi = x2 + 25m2 + 25o2 (mod 77).

From (8.15) and (8.28), we obtain

(8.29) /5 ee -(x2 + 25M2 + 25tr)/10(xw + 5mo) (mod 77).

Using (8.29) in (8.21), we get, appealing to (8.15),

(y)-l+y-'f) h,).\ f I L ö(xw + 5mü)

As both sides of the congruence (8.30) are integers (mod p), and since x, x2 — 125w2

and x + 5uv/w are independent of the choice of solution (x, u, v,w) of (8.15),

(8.30) holds mod p. Similarly, using J5(2,2) in place of J5(l, 1), we obtain an

analogous congruence to (8.30) for (3^). These congruences are due to Emma

Lehmer [23, p. 69]. Summarizing, we have

Theorem 8.1. Ifp = 5/+ 1 is prime and (x, u, v,w) is any solution 0/(8.15), then

(Y)5lL+"-y-'25-;>) (mod,).\ / / 2 \ 4(xw + 5ho) /

(y-iL.-'y-■*■?> (mod,».\ // 2\ 4(xw + 5mo) / V ^7

The next corollary follows immediately from Theorem 8.1. It was recently

rediscovered by Rajwade [34].

Corollary 8.1.1. Ifp = 5/ + 1 is prime and x is given uniquely by (8.15), then

x+(y) + (y)=o (mod/,).



9. e = 6. There are two representative binomial coefficients of order 6, namely

(2f) and (3/). In this section we establish a congruence for (2f) which, in conjunc-

tion with Corollary 4.1.1, gives

(9.1) I y = 2x (mod/7 = 6/+ 1).

We have been unable to find a reference to this result.

Consider the Jacobi sum 4(2,5). By (3.4) and a result of Jacobi [19, p. 69], we

have

4(2,5) = (-l)/4(5,5) = x¿'(4)4(4,4),

that is (by (3.7))

4(2,5) = X3'(2)4(2,2).

Since x3(2) ee 2(p~ l)/3 (mod 77) from (4.7) and (6.3) we obtain

'A (mod 77), if ,4 = 5 = 0(mod2),

4(2,5)-]r(A + 95) (mod 77), if A =5 = 1 (mod2)M = 5(mod4),

-Ua -95) (mod 77), iíA=B=\ (mod2),/4 = -5(mod4).

Thus by Theorem 5.1 we have

Theorem 9.1. Ifp = 6/ + 1 is prime and A, B are defined by (4.23), then

'(-\)f+lA(modp)

(]_

2

J_2

2/

/

if A ee 5 = 0 (mod 2),

f-(A + 95) (mod p) if A = B = 1 (mod2),,4 =5(mod4),

(-\YUa -9B)(modp) ifA=B = \ (mod2),/l ee -5(mod4).

Appealing to (4.6), we obtain

Theorem 9.2. Ifp = 6/ + 1 is prime and x, v are defined by (4.4), then

2A -/

2(-\)fx(modp) ¡/>EE0(mod3),

(-\)f(-x + 3v) (mod p) ify = \ (mod3),

(-l)f(-x-3y)(modp) if y = 2 (mod 3).

Example. We illustrate Theorems 9.1 and 9.2 by taking p = 991, so that/= 165,

x = 22,v= 13, ,4 =61,5 = 3. We have

~(\f5) =914= -17 (mod 991),

(_l/(_x + 3v) = (-l)f\(A -95) = -17.



10. e — 1. There are four representative binomial coefficients of order 7, namely,

By (2.2) and Lemma 2.1, we have

so that it suffices to determine

m - (rmodulo/7. In order to do this by means of Theorem 5.1 one must consider the Jacobi

sums 4(2,6), 4(3,6) and 4(4,6), respectively. Of these, 4(3,6) is an integer of the

subfield Qfñ of g(f7), as

a2(4(3,6)) =4(6,5) =4(3,6),

and we are able to reprove Jacobi's result [19] for (3f) (mod p) using Theorem 5.1.

The other two Jacobi sums are related to 4(1,1) by

4(2,6) = 4(6,6) = a6(4(l, 1)), 4(4,6) = 4(4,4) = o4(J7(\, 1)),

so that to determine (2/) and (*/) modulo p it suffices to consider 4(1,1). This

Jacobi sum, unlike 4(3,6), does not belong to a subfield of ö(f7). We are able to

express 4(1,1) in the form C,f7 + C2f72 + C3f73 + C& + G¿75 + Qf76 where the

C¡, i = 1,... ,6, are linear combinations of a non trivial solution (x,,... ,x6) of (4.17).

Using Theorem 5.1 we are able to obtain the congruence

If ) = -2(2x, + 7x55 + 2lx6S) (mod 77)

where

5 = f7 + tf - 2f3 - 2tf + f75 + tf, S = f7 - J72 - f75 + f76,

and 77 denotes any prime factor of p in the ring of integers of ô(f7), but,

unfortunately, we have not been able to determine 5 and S mod 77 in any aesthetic

form. Consequently, unlike the case e = 5, we are unable to give (2/) and (jf)

mod p explicitly in terms of invariants of the system (4.17), although a result

analogous to Theorem 8.1 (but more complicated) may well exist.

We are (in analogy to Rajwade's result [34]) able to evaluate

2¡H¡H>^-

First we show, however, how Theorem 5.1 can be used to deduce Jacobi's result [19].

The ring 5 of integers of ß(f7) is a unique factorization domain [27]. In 5, p

factors into primes as

(10.2) /7 = 77,772773774775776,



where 77 is any prime factor of p in 5 and 77, = 0,(77), i = 1,2,3,4,5,6. In precise

analogy to the case e = 5 (see 8.4) we may normalize 77 so that

(10.3) 77EEI (mod(l-f7)2).

By (3.12) we have

4(1,2) =0 (mod 77,772774)

St)

4(1,2) = wnxm2m4,

where u is an integer of 0(f7). In view of (3.5) we have mm = 1 so u is a unit of

0(f7). As all units of 0(f7) are of the form

(10.4) ±(ß + ß6)k'(ß2 + ß5)klß'

(see, for example, [36, p. 99]), it follows from (10.4) that £, = k2 = 0, therefore

(10.5) u=±ß', ¿ = 0,1,2,3,4,5,6.

But 4(1,2) = -1 (mod(l - f7)2) and 77,772774 ee 1 (mod(l - f7)2) so

MEE-l (mod(l -f7)2).

Thus (10.5) must hold with the minus sign and with 1 = 0, that is,

4(1,2) = -77,772774.

Next, as

02(4(1,2)) = a2(-77,772774) = -7727747r, =4(1,2),

we deduce that 4(1,2) £ 0(V-7~). Since 4(1,2) is an integer of 0(f7), it must be an

integer of Q(4-l), so there are integers X and Y with X = Y (mod 2) such that

(10.6) J1(l,2)=^(x+Ypî).

As 4(1,2)4(1,2) = p, by (3.5), we have

(10.7) 4/7 = X2 + 7y2,

which implies there exist integers x and v with X = 2x, Y = 2 v, and (from (10.6)

and (10.7))

4(1,2) = x +yfï, x2 + ly2=p.

As 4(1,2) = -1 (mod 1 — f7)2 and (as is easily checked),

fl = Í, + f72 - tf + tf - f75 - ?7 = 0 (mod(l - f7)2),

we have x = -1 (mod(l — f7)2) so x = -1 (mod 7).

Finally, using Theorem 5.1 we have

(10.8) ( y) = -4(3,6) ee -4(5,6) ee -4(172)

ee - (x — v/-7 j ee-2x (mod 77).


456 R.H.HUDSON AND K.S.WILLIAMS

As the quantitites in the congruence (10.8) are rational integers, we have the

following theorem due to Jacobi [19].

Theorem 10.1. Ifp = 7/ + 1 is prime and x and y are integers with p = x2 + ly2,

x = -1 (mod 7), then

- -2x (mod p).

We now show that a result analogous to that of Rajwade [34] follows easily from

Theorem 5.1 and the basic properties of Jacobi sums listed in §3.

First, note that a precisely analogous argument to the one above for 4(1,2) gives

4(1,1) =-77,774775 so 4(1, l) = 4(2,2) = 4(3,3) = 0 (mod77).

By Theorem 5.1 we have

(10.9) if ee -4(2,6) ee -4(6,6) (mod77),

(10.10) jyj ee-4(4,6) ee-4(4,4) (mod77),

(10.11) l42ff =-4(4,5) ee -4(5,5) (mod77).

Adding ( 10.9)-( 10.11 ) we obtain

(y)+(y)+(^)=-^7(M) (mod ,7).

Since

we have

6

2 4(', 0 ~ *i» x, ee 1 (mod7),

Theorem 10.2. If p = 7/ + 1 is prime and (x,,... ,x6) is a solution of (4.17) with

x, ee 1 (mod 7), then

IH'MZ+ , +U, ="*i (mod/7).

Example. We illustrate Theorem 10.2 by taking/? = 29 so that/= 4 and x, = 1

(see [47]). In agreement with Theorem 10.2 we have, for/ = 4,

(2f)+r/)+(2/) -i2+22+23^-] (m°d29)-

11. e = 8. There are four representative binomial coefficients of order 8, namely,

2/)'(T)-(4/) - \l



Now, by Corollary 4.1.2, we have

(11.1) (/H'^lv) (m°d^and appealing to Theorem 7.1, we obtain

Theorem 11.1. If p = 8/+ 1 is prime and a is given uniquely by p = a2 + b2,

a ee 1 (mod 4), then

(3/)-(-l)/+ft/42« (mod/7).

Next, from Lemma 2.1 and (2.2), we obtain

Thus from (11.1) and ( 11.2) we have

(11.3) (vH-^i4/) (m°d^-

Again, appealing to Lemma 2.1 and (2.2) we have

which gives, in view of (11.3),

(11.4) (2/)"(-1)/W4(4/) {m0dp)-

(11.3) and (11.4) show that it suffices to determine (4/) (mod p). In order to do this

we must consider 4( 1,4).

We set ß = fg = (1 + i)/ \¡2. The ring 5 of integers of 0(ß) is a unique

factorization domain [27], Let 77 denote a prime factor of p in 5. We have

«iH\ 4)i -IC\*\- G*^G*W - Gb(1)^(4) _ , .a3(4(l,4))-4(3,4)- - -4(1,4),

so 4(1,4) belongs in the subfield 0^-2 of 0(ß). As 4(1,4) is an integer of 0( ß), it

must be an integer of Q({^2). Thus we can set

(11.5) 40,4) = -(c + dfï),

where c and d are integers. As 4(1,4)4(1,4) = p, we have p = c2 + 2d2. Clearly,

we have

(11.6) I —-) - 1 ee 0(mod2) ûp\\-n.

Further, since 4-2 = ß(i + i), w^ have

(11.7) x8(«)-l=0(modf2) if(-) = +l,


458

(11.8)

R. H. HUDSON AND K. S. WILLIAMS

Xtt(n)-ß = 0 (modfl) if (^) = -1.

We now combine (11.6)-(11.8) and note that

p-\

2n=7

1 -n1,

2 in = 2

+ 1,

(ï)--

2(í-3), (¿| +1,

It clearly follows that

(11.9) 4(1,4) =-1 (mod2/^2),

c = 1 (mod 4).

so

(11.10)

Then by Theorem 5.1, we have

(4/)^(-l)/+'^(4T7y (mod*)

ee(_1)'+14(L4) (mod77)

ee(-1)/(c-í¿/^2) (mod 77).

But 4(1,4) EE 0 (mod 77) by (3.12), so c + df^2 = 0 (mod 77). Hence

(4/) ee^O^c (mod 77).

Thus we have proved

Theorem 11.2. If p = 8/ + 1 is prime with a and c defined uniquely by p — a2 + ¿>2

= c2 + 2d2, a = c = l(mod4),

(y)=(-r/2c (mod/7),

(2y)-(-D/+i/42c (mod/7),

(y)=(-l)"/42c (mod/7).

The first congruence in Theorem 11.2 is due to Jacobi [20] and Stern [40].



12. e = 9. There are six representative binomial coefficients of order 9, and using

(4.21) it is easy to show that all six are expressible in terms of

7M3;m5;)-In particular,

(12.1) (2/)S3('~1)/3(4/) (m0d/,)'

(12.2) jyj=3(,-.)/3J4/j {modpl

(12.3) (/H30,~1)/3(2/) {mOÚp)-

Unfortunately, we have been unable to determine any of these binomial coeffi-

cients explicitly. However, we are able to prove the following theorem analogous to a

result of Jacobi; see Theorem 14.1.

Theorem 12.1. Forp = 9/ + 1, 4/7 = A2 + 2752, A = 1 (mod 3), we have

(2/)(T)Proof. Since

4/1/5/1 / /3/\ _ 4/!5/!6/!

/(24//HM/(3/H <-»•

/(3/)f)\2f}< \fj 3/!3/!6/!'

the result follows immediately from (2.1) and (12.1)—(12.3).

13. e = 10. There are six representatives binomial coefficients of order 10, namely,

2/\ /3/\ I4f\ I5f\ I5f\ (6f\

(13-1) \fV\fV\fV\fV\vv\yvWe show that all of the binomial coefficients in (13.1) can be determined from the

lower order binomial coefficients (2^) and (62ff) which are given explicitly in Theorem

8.1.

We begin by taking the Davenport-Hasse relation (3.9) with e = 10, m = 5, t = 3,

to obtain

G.o(5)G,o(6) = x35(2)G10(3)G,0(8).

By (3.3) we have

40(5,8) = G,0(5)G10(8)/G,0(3) = X35(2)G10(8))2/G10(6)

= X35(2)4o(8,8) = x35(2)4o(8,4)

so, by Theorem 5.1, we have

5/\-

2/= (2(p-.)/5)3|4/j (modw))



where 77 is defined as in §8, so

5/| —(>i(p-1)/5\3| 4/

(13.2) \2f)={1(P~W5)\2f) {m0dp)-

From (4.10) and (13.2) we have

(13.3) (4/)s(-i/(2('",)/s)4(i/) (™dpy

Applying Lemma 2.1 (with g = 5, h = 2, k = 4) we have

<»*> (SKOVß) 0-,).Using (13.2) in (13.4) we obtain

(13.5) (y)3^1^"1^!^) (mod/,)'

Applying Lemma 2.1 (with g = 3, h = 1, k = 4), using (2.2), (13.3), and (13.5), we

obtain

(13.6) (3/H2<,,~,)/5)3(26/) {m°dp)-

Next, applying Lemma 2.1 (with g = 3, h = 1, A: = 5), using (2.2), (13.4)—(13.6), we

get

(13.7) (y)=2(,-0/5J6/j {modp)

Finally, applying Lemma 2.1 (with g = 2, h = 1, k = 5) and using (13.7) we obtain

(13.8) (yj^-l)'^*-1'/5)2^) (mod/')-

Combining (13.2), (13.3), and (13.5)—(13.8), we have the following new theorem.

Theorem 13.1. Let p = 10/+ 1 ¿>e a prime and let (x, u, v, w) be a solution of

(8.15). If 2 is a quintic residue of p (equivalently, x is even), we have

2/U(_1)/(3/)s(_I)/(V)s(1)//6/

x w(x2 - \25w2) 1H); ~T - ,/ ...... „/ (mod/7),

' \fl v

<x2- 125w2

8(xw + 5mü)

// ' \2f)

x w(x2 — \25w2

2 8(xw + 5 mu)

jc h>(x2 - 125w2) 1= (-0 ~T+ 0/-..-L.C ,\ (mod/7).



// 2 is a quintic nonresidue of p (equivalently x is odd), we can choose a solution

(x, u, v, w) of (8.15) satisfying u = 0 (mod2), x + u — v = 0 (mod4) so that (see

Lehmer [21])

2{p~X)/s=a(x,u,v,w) (mod/?)

where a = a(x, u, v, w) is given by (4.13). 77ie« we have

-1 t\f i( -x w(x2 - \25w2) \ . .= H « h-7T,-, , x (mod/7 ,

\ 2 8(->ctv + 5wt)) /

-(-■Mf+y"+'fi)) (mod")-\ ^ 8(xw + 5md) /

We close this section with two examples illustrating Theorem 13.1.

Example. Let p = 151 so that/ = 15 and 2 is a quintic residue of p. Then

(5)-(S)-(5)-(3)-» <-d'5"'

(S)-(S)-(SMS)-* ^151>-A solution (x, u, v, w) of (8.15) with x even is given by (x, u, v, w) = (-4,2,2,4).

In agreement with Theorem 13.1, we have

, ,.// x w(x2-\25w2)\ -4 4(16-2000) „ , J1C1.(-l)[-2- Z(xw + 5uv)) = T+ 8(-16 + 20) ~52 (m°dl51)'

and

// x w(x2- \25w2)\ _ -4 4(16-2000) _(-1M"2+ 8(xw + 5M,) J-T" 8(-16 + 20) = 95 (m°dl51)-

Example. Let /7 = 11 so that / = 1 and 2 is a quintic nonresidue of p. Note that

a = 4 so a2 EE 5 (mod 11), a3 = 9 (mod 11), a4 ee 3 (mod 11). Now it is easily

checked that

(îMîM.Mî)-* <>«*»>and, similarly,

(iM!MM)-< *-»>■Moreover, solutions of (8.15) are

(x,u,v,w) = (1,0,1,1), (1,-1,0,-1), (1,1,0,-1), (1,0,-1,1).



The first of these solutions satisfies (4.12) (u = 0 (mod2), x + u — v — 0 (mod4))

and, in agreement with Theorem 13.1, we have

,,,/-,/ x w(x2-\25w2)\ ( 1 1 - 125 \ „ , JllX

(-1)V(-2 - ¡(xw + 5uv) r-5h - 8(ÏTO)) "2 (m0dU)

and

(-l)VU - y-1^) - -3Í-1 + ±=J* ) EE4 (modll).v ; \ 2 &(xw + 5uv) ] \ 2 8(1 + 0) / v ;

14. e = 11. There are ten representative binomial coefficients of order 11, namely

2/\ /3/\ /4/\ /5/\ /6/\

i/n/n/n/n/4/\ /5/\ (6/\ /6/\ llf\

2/)'U//'\2//'l3/)'U//'It appears to be difficult to determine any of these explicitly modulo p in terms of

the variables of a quadratic partition of p such as

(14.2) 4/7 = a2 + 11¿>2, a = 2 (mod 11),

or the representation given in [25]. We first show that Theorem 5.1 can be used to

reprove a theorem of Jacobi [19] relating (3/), (\ff) and (3^) modulo p.

Let 77 be a prime factor of p in the unique factorization domain 5 of integers of

ô(fn). By Stickelberger's theorem (3.12), we have

4,(1,2) ~ 77,77277477677g, 4l(2,2) ~ 77,77277677777g, 4l(3,3) ~ 77,7r3775777779,

where, if a, and <x2 are integers of ß(fn), a, ~ a2 means that a,/a2 is a unit of the

ring of integers of ô(f,,). Hence,

(14.3) Y =4,(1,2)4,(3, 3)/4,(2,2) ~ 77,773774775779,

showing that y is an integer of ô(fM). Next, appealing to (3.3), we have

Y = G„(1)G„(3)G„(4)/G„(2)G1,(6),

so

o3(y) = G„(3)G„(9)G„(1)/G,,(6)G„(7).

Since, by (3.6), G,,(2)G,1(9) = G,,(4)G,,(7) =p, we obtain a3(y) = y, which shows

that y belongs in the subfield Ô(/-ÏT) of ô(f,,). As y is an integer of Q(ÇU), it must

be an integer of ß(V-ll ), and so has the form

(14.4) Y = -^(û + ^/IÎT),

where a, ¿7 are integers such that a = b (mod 2). From (14.3) and (3.5) we have

YY = P- Hence a and ¿7 satisfy the equation given in (14.2). The congruence in (14.2)

follows as

a = a + bpîï = -2y = 2 (mod(l - ?,,)2),

by (3.8).



Finally by Theorem 5.1 we have

4,(9,10) = - (y) (mod 77),

4,(8,8) = -(^) (mod 77),

4,(9,9) ee- J^j (mod 77),

so

(14.5) [^][\^/[\f^-y=$a-bfñ) (mod 77).

But from (14.3) and (14.4) we have

(14.6) Ua + bfÄ$ =0 (mod 77).

Hence from (14.5) and (14.6) we have

7)(S)and so appealing to (2.2), we obtain

Theorem 14.1. Ifp = 11/+ 1 is prime and a is defined uniquely by 4 p = a2 + 11¿>2,

a EE 2 (mod 11), we then have

mThis is equivalent to Jacobi's result [19]

1

r)/ I 2 J =a (mod 77),

/(î/) =fl (mod^)'

"-/!3/!4/!5/!9/T (m°d/,)-

Example. With/7 = 89, so that/ = 8, ü = -9, ¿7 = 5, we have

/24U48\ //32\ _(64)(72) _ 79 _I 8 JI24J/ I 16 j =—22— =TT = "9 (mod89)-

15. e = 12. There are eight representative binomial coefficients of order 12,

namely,

We show that all the binomial coefficients in (15.1) can be determined from the

lower order binomial coefficients (\ff),(%) and (4^).

We begin by determining (3^) in terms of (\ff) modulo p. Let 5 be a prime ideal

divisor of p in Q(Çn) and define g and Xn as m §3- Then it is known (see, for

example, Whiteman [44, p. 61]) that 42(3,3) = -a + ¿7/ where p = a2 + ¿>2, a = 1

(mod 4).



Appealing to Whiteman's cyclotomic numbers of order 12 [44] we have

(15.2) 42(l,2) = (-l)/cX,2(3)42(2,4),

where c is given by

(-\)f ifa = 1 (mod3),¿7EE0(mod3),

if a EE2(mod3),¿>EE0(mod3),(15.3)

(-1)/+!

(-!)'/ if 6= 1 (mod3),flEE0(mod3),

/;(-l)'i if ¿7 = 2 (mod 3), a = 0 (mod 3).

Now 3 is clearly a quadratic residue of p so that x*Q) = (-1)7 if ¿> = 0 (mod 3)

and X4(3) = (-l)f+i if a = 0 (mod 3) (see, for example, [22, p. 24]). Taking con-

jugates on both sides of (15.2) we have

42(10,11) =ec42(10,8),

where

+ 1 if¿7 = 0(mod3),

-1 ifûEE0(mod3).

Appealing to Theorem 5.1, we have

(-i)/+'i10/) = -r;;i (mod5).

Finally, as 42(3,3) = -a + ¿7/ ee 0 (mod P) we have, using (2.2),

(15.4)

10/

4/

3/Mv Mwhere

(15.5)

(-D1(-D/+1

(-i)V«

if a ee 1 (mod 3), ¿7 EE0(mod3),

if a ee 2 (mod 3), ¿7 EE0(mod3),

if¿>EE 1 (mod 3), a =0(mod3),

\/+1(-1); ¿7/a if ¿> ee 2 (mod3), a = 0 (mod3).

We now show that the 7 remaining binomial coefficients of order 12 may be

determined in terms of lower order binomial coefficients.

Corollary 4.1.5 relates (5/) and (ff) modulo p. However, Corollary 4.4.1 gives a

simpler congruence, namely,

(15.6)5/

/

8/

4/(mod p ).

Corollary 4.2.2 gives the congruence

6/\ .. / 6/

/)=£U



Corollary 4.3.1 gives the congruence

(15.7) (y) =(2"-»/3)a(y) EEe(2^-')/3)2|6/j {modp)

Appealing to (2.2) and Lemma 2.1 (g = 2, h = 1, k = 9) we have

(y)/(7)-<-'>i7ra-M/M <-»•Thus, using (4.3), we obtain

(4/)H2/)(r/)/(3/)-"-'i2/)(26//)/(3/)<-^

Now using (15.4) and (15.7) we have

(15.8) (TlHv) (m°d/,)'Next using Lemma 2.1 (g = 7, ¿¡ = 3, k = 4) we have

so, using (15.5) and (15.8), we obtain

(15.9) (vH'^'lv) (m°d/,)-Again appealing to Lemma 2.1 (g = 6, A = 3, k = 5), we have

($/(£H-0/(f/) *-')■Using (15.9) we have

(15.10) (25/Hr'(2/) {m0dp)-

Finally, appealing to Lemma 2.1 (g = 4, h = 2, & = 5) and using (2.2), we have

($/&)-<-<)/(!HS)/(S) *-'>■Using (4.3), (15.9) and (15.10) we have

so that after cancellation we obtain

(15.11) (27/)^(-1)/2<'")/3e(3/) (m°d/,)-


466 R H HUDSON AND K S. WILLIAMS

Combining (15.4) and (15.11) and appealing to (4.5), Corollary 4.1.1, and Theo-

rems 6.1, 7.1 and 9.2, we have

Theorem 15.1. Let p = 12/+ 1 = a2 + ¿>2 = x2 + 3y2 be a prime with a = 1

(mod 4), x ee 1 (mod 3), and let 4p = A2 + 21B2 with A = 1 (mod 3). Then we have

the following congruences modulo p:

(y)--*- (3;h- (i)-**- (5;m(6/h- •2(-\)fe2<t,a. (^)=2(-l)'fc,,

/5/\ 2x lip

U/j " 0 ' \2f)where 0 is given by (15.5) and $ by

1 ify = 0(mod3),

(x + 3y)/(x-3y) ify = 1 (mod3),

(x - 3y)/ (x + 3y) ify = 2 (mod3).

4>

Example. For p =£ 97, formulas (15.4)—(15.11) and Theorem 14.1 can be easily

checked from the following brief table of values. (See Table 2.)

16. e = 13. Since a,(43(l,3)) = 43(3,9) = 43(1,3), 43(1,3) is an integer of the

field g(//26 + 6/ÎT ). Zee [15, p. 263] has shown that

(16.1) 43(1,3) = ^lx + w{\3 +/íi/y/26 + 6/Í3" + t>^26 - 6/Ï3 )),

where (x, u, v, w) is a solution of the system

16/7 = x2 + 26«2 + 26t)2 + 13w2, x = 9 (mod 13),(16.2)

^xw = 3v — 4uv — 3w

We prove

Theorem 16.1. If p = 13/+ 1 is prime then

4/\ x 3(x2- \3w2)wf}=-^ + ~tt-, n \ (mod/7)/ / 2 8(xw + 13«ti)

and

lf\ _ x 3(x2- \3w2)w

2fj 2 8(xw + \3uv)

where (x, u, v, w) is any solution of (16.2).

(mod/7),

Proof. The ring of integers of Q(Ç]3) is a unique factorization domain (see, for

example, [27]). Let 77 be a prime dividing p in Q(${3). By Theorem 5.1 and (2.2), we

have

(16.3) (yU(1°/)=--UlO,12) (mod 77).



Since, by (3.12), we have

(16.4) 43(1,3)=0 (mod 77),

adding (16.3) and (16.4) and appealing to (16.1), we obtain

29U(43(l,3)) = -(y) (mod 77),

that is,

(16.5) I y \=-Ux + w{ñ) (mod 77).

For brevity we set ß = f ,3. We have (see, for example, [51, pp. 262-263])

(16.6) ß + ß3 + ß4 + ß9 + ß10 + ß12 = |(/Ï3 - l),

(16.7) ß2 + ß5 + ß6 + ß7 + ß8 + ß" = ^(-i/l3 - 1 ),

(16.8) ß2 + ß5 + ß6 - ß7 - ß8 - ß" = I/26 + 6/Ï3 ,

(16.9) ß + ß3 - ß4 + ß9 - ß10 - ß12 = ^26 - 6/Î3 .

By (3.12) and (16.1) we have

(16.10) jíx + w/Ï3 +»Îm^26 + 6/Ï3 + o^26 - 6/Ï3 )) =0 (mod 77).

Applying the automorphism a2 to (16.10) and appealing to (16.1)—( 16.9), we obtain

(16.11) t(jc- w/Ï3 -ií«j/26-6/iy -t>^26 + 6/Ï3 )) ee 0 (mod 77).

From (16.1) and (16.4) we have

(16.12) \(x + w{Ï3 + iíu^J 26 + 6{¡3 + t^26 - 6/Í3 )) ee 0 (mod ir).

Adding (16.11) and (16.12) we get

(16.13) 2* + ¡«(^26 + 6/Î3 - ^26 - 6/Ï3 )

+ i«í|/26 + 6/Ï3 + ^26 - 6/Ï3 ) ee 0 (mod 77).

Taking the term 2x to the right-hand side of (16.13) and squaring, we obtain, after

some simplication,

(16.14) /l3 =(x2+ \3u2+ \3v2)/(2u2 - 6uv - 2v2) (mod 77).

Next, using (16.2), we get

(16.15) /Ï3 ee -3(x2 - \3w2)/4(xw + 13no) (mod 77).



Substituting (16.15) into (16.5), we have

4/\ _ x 3(x2- 13w2) w(16.16) ; = -- + -¿-——^ (mod 77).

\ / / 2 8(xw + 13mu)

R. J. Evans (personal communication) has shown that all solutions of (16.2) are

given by

(x, u,v,w), (x, -u, -v, w), (x,v,-u,-w), (x,-v, u, -w).

Hence, x, w2 and uv/w are independent of the choice of solution of (16.2), and thus

(16.16) holds (mod p). This completes the proof of the first part of the theorem. The

second part follows similarly, by considering Jl3(l, 11) = Jn(l, 8) = a7(43(L 3)) in

place of 43(1,3).

Corollary 16.1. If p = 13/ + 1 is prime and x is given uniquely by (16.2). then

4M ¡If// 12// (mod^)-

Example. We taken p = 53 so that /= 4. A solution of (16.2) is given by

(x,u,v,w) = (9,3,4, -3), so that

-x/2 = 22, 3(jc2- 13h>2)w/8(;cr>+ 13m?) ee 49 (mod53).

Hence, by Theorem 6.1, we have (mod 53),

' ee22 + 49ee 18, [Ja

Indeed, we have

/ ' 12/

4/\ _ /16 1820 = 18 (mod 53),

7/

2/i288) = 3108105 =26 (mod 53).

17. e = 14. There are 16 representative binomial coefficients to consider when

e = 14 (see Table in §2), four of which are of lower order. The binomial coefficient

(%) is given by Theorem 10.1. In this section we show that the 12 representatives of

order 14 can all be expressed in terms of the lower order binomial coefficients

\'m - (si-In particular, we prove

Theorem 17.1. If p = 14/+ 1 is prime then the sixteen representative binomial

coefficients can all be expressed in terms of the lower order binomial coefficients



We have (mod p ),

(2/i=2'"-"/'i4/) =(-i)'2«'-""(y)

E2"""/i'3ä("l)'2,r""('/)'

(3/) s(_1)/2«,-„/,(y) S(V| s22,,.,)/7|9/j s2„„,)/7|4/j

('/)-*--(*) ,(-,»—(^-(^-^"i?)-

Proof. We begin by noting that

(ï)/(ï)-(y)/(nFor brevity we denote f ,4 by ß.

From the work of Dickson [9] (see also Muskat [28]) we have

(17.2) 44(l,4) = ß8'nV2,44(4,4) [28,(4.7)],

and

(17.3) 44(l,6) = ß12'nV2)44(6,6) [28,(4.8)].

Applying the automorphisms a,3 and a,, to (17.2) and (17.3), respectively, we obtain

44(l3,10) = ß6inV2,44(10,10)

and

44(n,io) = ß6indg<2l/,4(io,io),

so

44(i3,io) = 44(n,io).

Hence by Theorem 5.1 we have

(7)s(v)(mod7r)'

where 77 is a prime ideal divisor of P in Q(ß). Appealing to (2.2), we obtain

o™) (5/Hv)(mod^-

The proof of

(17.5) (/H*/) (m0dp)

is similar.


470 R. H HUDSON AND K.S.WILLIAMS

Next, appealing to (3.11) with n = 2 and / = 1,2,... ,m - 1 we have, using (2.2),

2'<-^=(-ir(2/)/(7/) <mod^-

Applying Lemma 2.1 with g = 2r, h = t, k = m, e = 2m, we have from (17.1),

Appealing to (17.2) and (17.3) with ? = 1,2,3, and m = 7, and using (2.2) we

obtain (mod /7 ),

(17.6) (2/)S(-d'2«-'>/'(7/). (y)^*'-""^).

(17.8, (3;)S(-»'2-—(3v), (*)-*"->"($.

Moreover, from (4.16) we have

(17.9) (y) =(-\)f2^-^ 0/7/ 7/

2/

Theorem 17.1 now follows easily from (17.1)—(17.9).

Remark. The congruences in (17.4), (17.5), and

(*)-(-./(5) <-,)(Theorem 17.1) are of Cauchy-Whiteman type (see [17], (1.6), and §21).

18. e = 15. There are 19 representative binomial coefficients to consider when

e = 15, including 3 of lower order. We begin this section by establishing seven

congruences relating these representatives solely by powers of 5ip~l)/3 or 3<-p-V/5.

We prove the following.

Theorem 18.1. Ifp = 15/ + 1 then we have the following congruences (mod p):

TH"-"" (l\3/1 = 3(P-I)/5J 6/j — 32(/7-l)/5[ 7/\ _ 33(/,_1)/5/ 7/

á-*"■&)■ (r)-"-"i3Ív



Proof. The first and fourth congruences in Theorems 18.1 are exactly (4.26) and

(4.27).The third congruence can be established using (2.2) and Lemma 2.1 with g = 3,

h = 2, k = 6 to obtain

the second congruence follows at once from this, and the third follows from the

second.

The fourth, sixth and seventh congruences are easy consequences of (2.1) and

(3.11). For, using (3.11) with « = 3, t = 3, m = 5, it follows from (2.1) that

33<,-„/s=9/!10/!15/!=J7AM_.2¿!¿!3/! 8/! 13/! /!6/! 3/! lIIluu^-

Similarly, using (3.11) with n = 3, t = 2, m = 5, it follows from (2.1) that

32(,-l,/,=6/!10/!15/!=_g¿!_,VW! {mod p)2/! 7/! 12/! 2/! 6/! 9/! V™»^-

Next, using (3.11) with n = 3, t = 1, m = 5 it follows from (2.1) that

/!6/!ll/! /!3/! 6/! KllûP)-

Finally the fifth congruence clearly follows from the second and the fourth.

The last two congruences in Theorem 18.1 are particularly interesting because they

relate binomial coefficients of order 15 to the lower order binomial coefficients given

explicitly in terms of the system (8.15) in Theorem 8.1. In particular, we have

(18.1) f4/)EE3^-Û+y-123w;M (mod/,)v ' \ f } \ 2 S(xw + 5uv) ] v Fi

and

(18.2, (2*/)s3^(_f-Öî) (mod,,

Example. Let/7 = 31 = 15(2) + 1 so that (x, u, v,w) = (11,1,-2,1) is a solution

of (8.15). As 36 EE 16 (mod 31) and 313 = 8 (mod 31), we have

(«)=28ee16(-^-|)ee_96 (mod31),

(146)EE22EE8(-y + |)EE-40 (mod31),

in agreement with (18.1) and (18.2).

Next we use the Jacobi sum 45(1,4) to explicitly determine (s/) and (7/f)m terms

of parameters in the quadratic forms p = g2 + 15/i2 and 4p = A2 + 21B2. In

particular we prove



Theorem 18.2. Let p = 15/+ 1 = g2 + \5h2, 4p = A2 + 27ß2 EE l

(mod 3). 7/7ie« we have

2g(modp)

5/

/

2^1g-18ßg

A +9B

2AG + 18ßg

9ß

;/,4ßEE0(mod5),

(mod p) ifA=Bor~2B (mod 5),

(mod/?) if A EE -B or 2B (mod 5),

7/

2/

2g(mod/7) z//lßEE0(mod5),

2^f - Q»gg (mod P) <M = Bor ~2B (mod5),

2^g~18ßg

/l +9ß(mod/7) //^ EE -ß 07-25 (mod 5).

Proof. From Muskat [30,p. 498] we have 45(1,4) = 5(p~u/3(-g + h^f\5i). Ap-

pealing directly to Lemma 6 of [45] and noting that

45(14,ll) = (5<'-"/3)2(-g-/r/Ï57),

one immediately deduces (by adding and using (4.28)) the first congruence in

Theorem 18.2.

The second congruence is then an immediate consequence of (4.27), completing

the proof of Theorem 18.2.

Consider now the Diophantine system

16/7 = x2 + 50m2 + 50u2 + 125w2, x = 1 (mod 5),

^xw — v2 — 4uv — u2\

let a = 1 if 3<-p~1)/s = + 1, and if 3 is a quintic nonresidue of p, let

(18.3)

^(125v x2) + 2(xw + 5mü)(25w - x + 20m - 10u)(18.4) a(x, u, v, w) =

w(125w2 - x2) + 2(xw + 5«u)(25w - x - 20w + 10u)

where (x, u, v, w) is the unique solution of one of

(a) x = l, u = \, o = 0, w EE 2 (mod3),

(b) x = 2, «ee2, d=0, w ee 1 (mod 3),

(c) x = \, mee 2, 0=1, w ee 1 (mod3),

(d) x = 2, «ee 1, v = 2, w = 2 (mod3).

Then Williams [49] has shown that

(18.6) 3</>-n/5 = a(x, u,v,w) (mod p).

A straightforward calculation shows that

(18.7) (3(^1)/5)2EE«(x,-t;,M,-w) (mod/7),

(18.8) (3{p~i)/5f = a(x,v,-u,-w) (mod p),

(18.9) (3(p-[)/5)4 = a(x,-u,-v,w) (mod p).



Using Jacobi sums first given by Dickson [10] with a sign ambiguity, and later by

Muskat [30] with the sign ambiguity neatly removed, we obtain

Theorem 18.3. Let p = 15/+ 1 = g2 + 15/i2, g ee 1 (mod 3), and let (x, u, v,w)

be the unique solution o/(18.5) satisfying (18.3). 777e7i we have, modulo p,

j3/j ^(_1)t^]2ga(^)_IÍ)_ü)W)j |yj =(-\f^2ga(x,-v,u,-w),

(62ff) =(-\fg/5]2ga(x,v,-u,-w), (V) ^(-lf^2ga(x,u,v,w).

Proof. From Muskat [30, p. 487] we have

45(l,4) = i7(5<"-1)/3)2(3^-')/5)45(l,2),

where b = (-1)E2^/5J by [30, p. 498], Thus

45(14,11) = 05("-|)/3(3<"-|>/5)445(l4,13),

so that, again appealing to Lemma 6 of [45], we deduce that

(y) EEE(-l)[2^/5]5<"-,)/3(3^-1>/5)4iyj (mod/7)

from which the first congruence in Theorem 18.3 follows in view of (4.28), Theorem

18.2, and (18.9). Theorem 18.1 now gives the remaining congruences.

Example. Let/7 = 661 so that 3(p~l)/5 = 1 (mod661),

^-1)/3 = ̂ f = 1-364 (mod661),

(5<Jp-D/3)2 = 22/76 ee 296 (mod661),

g = -11 so that (-l)l2«/5l = -1. From Theorems 18.2 and 18.3 we have, for/= 4,

5f| (-22)(296) ee 98 (mod661),

2^) =(-22)(364) = 585 (mod 661),

and

"(7/)=(26/)S(37/H(-1)[2g/5,2^22 (mod661)>

all in agreement with values for these binomial coefficients obtained from computer

data.



The remaining binomial coefficients of order 15, namely

are related to one another in the following theorem.

Theorem 18.4. Let p = 15/+ 1. Then we have

6/

/ )/ >-«

V(8/ 4//(

5/

2/= a

9/

4/

l9fW//3/// I// U/

= (-l)[2g/5]a35<"-,)/3 (mod/7).

Proof. Theorem 18.4 follows immediately from Theorems 18.2 and 18.3 as

8/)

4/j

6/1/ )/(2ff)=iZ)AvA miv-w/lf)/(vf

4/W (5/VI' 12/

T)7/)/'7/2/ 3/

and, making use of (2.2),

9/

4/ )/8/

4/ ='-"f/)/(5/) = /(as/is even.

An explicit determination of the 8 binomial coefficients in Theorem 18.4 appears

to require the quadratic form discussed on p. 198 of [10], An easy computation

shows that each of these binomial coefficients may be determined given an explicit

determination for any one of them. The following theorem provides a determination

of (\ff) which involves only the forms p = g2 + \5h2, 4p = A2 + 27ß2, and 16/7 =

x2 + 50m2 + 50t;2 + 125w2, x = 1 (mod5), xw = v2 — 4mü — m2; this determina-

tion has a sign ambiguity in the form of a square root.

Theorem 18.5. Let p = 15/+ 1 = g2 + I5h2, 4p + A2 + 27ß2, A = g = 1

(mod 3), and let (x, u, v,w) be the unique solution of (18.5) which is a solution of

( 18.3). Then we have

5/

2/

(-2ga(x,-v, u,-w)y_y+/A)l/2 if AB = 0 (mod 5),

(-2ga(*, -v, u, -w)y_(A + 9B)y+/ (A2 - 9AB))V2

if A =B or-2B(mod5),

(-2ga(x,-v, u,-w)y_(A - 9B)y+/(A2 + 9.4ß))1/2

if A = -Bor2B(mod5),

where

y+ = y+ (x, u, v, w) = -x/2 + w(x2 — 125>v2)/8(xh> + 5uv),

y_= y_(x, m, t7, w) = -x/2 — w(x2 — \25w2)/cl(xw + 5mu).



Proof. We have

(8/

from which it follows

/(«)/(& (5M$-($/ß)«-'>

2ÍHM(^)/m <-»■Hence Theorem 18.5 follows from Theorem 6.1, (18.2), Theorem 8.1, and Theorem

18.2.

Example. Let p = 661 so that we may take (x, u, v, w) = (1,3,0, -9), -2g = 22,

A =49 ee -6 ee -2ß(mod5), and (/I + 9ß)/(^ - 9ß) ee 364 (mod661). Then

1 -9(1 - 125(81)) -1 + 113 ., -1-113y+ = - - +-g(Z9)-= " -1— =56' y = - — - = "57-

As a = 1, Theorem 18.5 gives

2^ ee /(22)(604)(364)(56)/49 =i/y- (mod 661).

Computer data gives

(2828°)ee325 (mod 661),

which clearly is in agreement as ((325)(7))2 ee -5 (mod 661).

19. e = 16. There are 16 representative binomial coefficients of order 16, namely,

2f\ /3/\ (4f\ I5f\ ¡5f\ (6f\ I6f\ llf\//'l//'\//,\//'\2//'\//'\3//'\/

V\ [V\ (8/\ I if] I9f\ ¡9f\ I9f\ /10/2/r\3/i'l//'l3/i'U//'\3//'\4/)'i 5/

We begin by noting the following congruences between binomial coefficients of

lower order; these are immediate consequences of (11.1), (11.3), and (11.4) respec-

tively. Throughout this sectionp = a2 + b2 = c2 + 2d2, a = c = 1 (mod 4).

(19.1) (f/H-^i!!/) <mod^

(19.2) (7f) ^h/4(lff) <mod^

(19.3) (í/H"1^'/) (m°d^-

Two of the above 16 binomial coefficients may be related to the lower order

binomial coefficients (f^) and (\Jf) as follows:

(19.4) (/H"1^"'^!'/){modp)'


476 R. H. HUDSON AND K S WILLIAMS

(19.5) (25/H-1)/2("')/8(2/) {m0dp)-

The first congruence is an immediate consequence of Theorem 4.1. To prove (19.5)

note first that from (3.11) we have

(1,6) ^,.im.(«/)/('y) (md„.

Next, using (2.6) with g = 11, h = 3, k = 6, we have

By (19.3) and (2.2) we have

(v)-^Ív) - WHv)from which (19.5) follows immediately as 2(p~i)/4 = (-\)b/4 (mod /7).

Now as 2(p~i)/8 = +1 or -1 modulo p according as b = 0 (mod 16) or 6 ee 8

(mod 16), we have the following theorem analogous to Theorem 11.2.

Theorem 19.1. Let p = 16/ + 1 = a2 + b2 = c2 + 2d2, a = c=\ (mod 4), be a

prime for which b = 0 (mod 8). Then

(-o'!y) ^í-1)^/) =2i-°'--2c (m°d^)

according as b = 0 (mod 16) or ¿7 ee 8 (mod 16).

When ¿7 = 0 (mod 8) we can use [23, (50), p. 70] to obtain

Theorem 19.2. Let p = 16/+ 1 = a2 + b2 = c2 + 2d2, a = c=\ (mod 4), be a

prime for which b Z 0 (mod 8). Then

/7/\_ /5/\ 2èc -2bc , . ,(^=-(2^=— or-^- (mod/7)

according as b = 4 (mod 16) or ¿7 ee 12 (mod 16).

Next, we establish 8 congruences which show that each of the remaining 14

representative binomial coefficients of order 16 are related to at least one other by

the quantity 2ip~i)/s. All such interrelationships are easily deducible from these 8

congruences. We first establish 8 congruences which relate

mm - (soto either (^) or (%), since for these two binomial coefficients we will give an explicit

determination in terms of the system given in [14] (see also [26, p. 366]).

(19.7) (2/)^(-l/2<'-'V«(y) (mod/7),



(19.8) (2/H2('~I)/8)3(8/) {m°dp)>

(19.9) (vH"1^2^078^!!/) (mod^>

(19.10) (yj^t-1^'"^'!?/) (mod^"

To prove (19.7) we use (19.4), (2.2), and (2.6). Taking g - 14, h = 8, k = 9, in

(2.6) we have

Since

by (2.2), the result follows at once from (19.4). Next taking g = 1, h = 1, & = 8, in

(2.6) we have

Wfh<%v) «-'>so (19.9) follows from (19.4).

Now (19.9) is immediate from (19.6) as

by (2.2). Next, taking g = 8, h = 3, k = 6 in (2.6) we have

(XHW)so (19.10) follows from (19.9).

Similarly, we may establish the following congruences:

(19.11) (!/)"2(^'>/8(6/) (m0dp)'

(19.12) (3/H2<'"')/8(4/) {m0dp)'

(19-13) (ïf) -^^^(ïf) {modp)>

(19.14) (yj^i-l/^-'^Jy) (mod/7).

Taking g = 1, h = 6, k = 10, in (2.6), we have



However,

by (19.2) so (19.11) follows from (19.4).

Taking g = 6, h = 1, & = 4, in (2.6) we have

6f\(\0f\ _(4f\l\2f//\3//-\//\5/< (m°d^

](^-<%4//) ^

so ( 19.12) is an immediate consequence of ( 19.11 ), noting that

(•v)-(-,^(ni-„-(»/)-(../(:;)(-„

by (2.2).

Taking g = 11, ß = 9, k = 12 in (2.6) we have

so (19.13) follows from (19.5), noting that

(■¿1-(£)<-" «d (^n-o'íi;).™-,)by (2.2).

Finally, taking g = 13, /i = 11, k = 12, in (2.6) we have

(ni)(3/H-'>tiM *-')so (19.14) also follows from (19.5), noting that

(^-(-i/^O-,)-!^)-^'/)^,

by (2.2).

Before proceeding to our determination of (^) and (3^) we wish to note that all

congruences between representative binomial coefficients of order 16 of Cauchy-

Whiteman type (see [17], (1.6), and §21) are readily deduced from the above

congruences. Computer data shows that the only such congruences are

(19.15) (vH^'iv) (m°d^'

(19.16) O1) ̂ -l)í+b/Í92ff) {m0dpl

(19.17) (/)"(-1)6/4(25/) (m°d^-

From (19.9) and (19.10) we have at once (19.15), and from (19.7) and (19.8) we

have (19.16). Finally, (19.17) follows from (19.3)-(19.5).



Now, appealing to (2.2) and Theorem 5.1, we have

46(15,9)ee-(8//) (mod 77),

where 77 is a prime ideal divisor of p in Q(t¡,6),

(19.18) p = x2 + 2m2 + 2v2 + 2w2, 2xv = u2 - 2uw - w2,

x ee 1 (mod 8), m ee u ee w ee 0 (mod 2). For the duration of this section we let

S — ?16-

From (3.5) and [43, p. 405] one obtains

46(i5,9) = (-i)'[-* - »a2 - n - u(t+f7) - w(?3 + ni

It is easy to see from [26] (see, e.g., [18, (3.20)] with x ee -1 (mod 8)) that

(19.19) -x - ü(f2 - f6) + m(£ + f7) + w(f3 + f5)EE0 (mod 77)

and

(19.20) 2v2-x2=(u2-w2 + 2uw)(Ç2-Ç6) (mod 77).

Thus

(19.21) (8//)EE-46(15,9)EE2(-l)/+'{-x-ua2-f6)} (mod 77)

ee 2(-\)f{x - v(x2 - 2v2)/ (u2 -w2 + 2uw)} (mod 77).

As the expressions on the left and right of (19.21) are rational integers, the

congruence holds (mod p ).

Similarly (mapping 6 -» 63) we have

/„(».H) = (-\)f{-x + »u2 - f6) - u(f3 + f5) + „(s + r)}.

Moreover, (19.19) becomes

(19.22) -x + v(K2 - f6) - M(f3 + ?5) + w(t + f7) ee 0 (mod 77).

As

4,(13,11) s-I ^j (mod 77)

from Theorem 5.1 we have

(19.23) (^)EE-46(13,ll)EE2(-l)/+'{-x + t;a2-^)} (mod77)

ee 2(-\)f{x + v(x2 - 2v2)/(u2 - w2 + 2mw)} (mod 77),

so, as before, this congruence holds (mod p ).


480 R H HUDSON AND K. S. WILLIAMS

Combining (19.7)-(19.10), (19.21), and (19.23) we have

Theorem 19.3. Let p = 16/+ 1 = a2 + b2 = x2 + 2m2 + 2v2 + 2w2, 2xv = u2

— 2uw — w2, with signs chosen so that a ee 1 (mod4), x = 1 (mod 8), and u = v = w

ee 0 (mod 2). 77ie7? we have

^/8/\ -l2f\ =™L_ v(x2-2v2) 1 =/ n/û2/9/(-i^i; = ; ^-/.wa+2^ -h)^2» (™d'>

v/fl|8/\ _/l0/\_,,i t;(x2-2t)2) 1 _û2/6/

w/îere 0 ee +1, -6/a, -1, or +6/a (mod p) according as b ee 0,4,8 or 12 (mod 16).

Example. For /7 = 113, /= 7, o = 8, (jc, m, u, w) = (1,-6,4, -2). In agreement

with Theorem 19.3 we have

('ÍHvHvH™ ̂ -^)-«'(—»)■The remaining 8 representative binomial coefficients of order 16 we are only able

to determine up to sign. Let

5, =8,(x, M, ü,w) = 2(-1)/{jc - u(x2- 2u2)/(m2- h>2 + 2mw)},

o2 = 52(x,M,ü,w) = 2(-l)/{x + u(jc2-2u2)/(m2- w2 + 2uw)}.

Then we have

Theorem 19.4. Let p = 16/ + 1 = a2 + b2 = c2 + 2¿2 = x2 + 2m2 + 2u2 +

2w2, 2xt> = m2 — 2uw — w2, with signs chosen so that a = c = \ (mod4), x ee 1

(mod 8), m ee v ee w ee 0 (mod 2). 77ie« we have the following congruences (mod p):

1/2 / 7A / i \ 1/2

4/)=(^)'/2- (^-(•Af)" ($-«-'>-.Aif.

'y)-((-ir«Af)w'.

Proof. In view of (19.11)—(19.14) it suffices to prove the congruences for

zmw- - r;License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use


To prove the congruences for (Yf) and (3/) we note that

(M-(X)- mH'tUand

so that appealing to Theorem 11.1, (19.4), (19.13), (19.14) and Theorem 11.2 we have

(mod p throughout)

7/\2_/6/W7/W8/\/3/\ //5/W8/1

2// \2//\//\3//\/// \fj\fj

EE(-l)V42a(2("-|»/8)(2cô2)(2<^"/8)3/ô|,

and

:3/r_/7/W6/\/7/\/8/\ //7/\/8/

/| l//\2//\2//\/)/(3//\3/j

EE(-l)/2("-|)/82c(-l)fe/42a(-l)/(2("-,,/8)3ÔI/ô2.

To prove the congruences for (4i) and (6/) we use Lemma 2.2 and (19.12) to see

that

.'4/\/6/\ (-1/0,02(19.24)

f !\ f I " 2('-,)/8

and we use (19.5) and Theorems 11.1 and 11.2 to see that

<-> (DAIHvViZ)= (-l)/2(p-"/8(-DV42c_, lV/2(,-,v.£

(-l)"/42a ^

From (19.24) and (19.25) the stated congruences for (^) and (6/) follow at once.

The next theorem taken in conjunction with ( 19.11)—(19.14) shows that a correct

sign determination for one of the congruences in Theorem 19.4 suffices to fix the

sign for the remaining seven.

Theorem 19.5. Let p = 16/ + 1 = a2 + b2 = c2 + 2d2, a ee c ee 1 (mod4). Then

we have

)(^)-4ac (mod/7), (4J]/( 9/f) =(-!)'£ (mod/7).5/)(V

Proof. The first congruence in Theorem 19.5 follows by combining (19.13) and

5/\/7/\ _ (6f\llf)-(-l)f4ac(Vp-^*f (mod/7)



(see (19.4), Theorems 11.1, 11.2). The second congruence follows at once from

(19.11) and (19.25).

We close this section by noting that results similar to Theorem 19.1 may be

deduced for e = 32 and e = 64, making use of the system given by (19.18), as

2</>-i)/i6 an(j 2</'~i>/32 have recently been determined (see Evans [12] and Hudson

and Williams [18]) in terms of the parameters in this system and those in Theorem

19.1.

20. e = 20. For p = 20/ + 1 there are 24 representative binomial coefficients of

order 20 and 9 lower order representatives. We begin this section by showing that 10

of the 33 binomial coefficients of order 20 may be expressed in terms of the

parameters in the representations p — a2 + b2 = e2 + 5/2, a=\ (mod4). In

[45,Theorem 3] Whiteman proved that for p = 20/+ 1 = a2 + b2 = e2 + 5/2,

a ee 1 (mod 4), we have

and according as b = 0 (mod 5) or b ee 0 (mod 5),

(7Mv)-P <-"•resolving the ambiguity in the congruence of Cauchy [5, p. 37]. If b = 0 (mod 5) then

(as the sign of a is determined by the condition a ee 1 (mod 4)) Whiteman showed

that e may be expressed unambiguously by the condition e ee a (mod 5), so for

p = 20/ + 1 = a2 + b2 = e2 + 5/\ o ee 0 (mod 5), we have

(20.3) My ee j ^ I ee 2e (mod/7) (a ee 1 (mod4), e ee a (mod5)).

The sign of b is not fixed. However, comparing formulas (4.7) and (4.13) of [45]

one sees readily that e may be expressed unambiguously by the condition e =\b\

(mod5). (Set e = (-\)fb' (where b' denotes Whiteman's b) when 5 \b; the determina-

tion in (20.4) requires choosing a primitive root g with gi{ = a/\b\ (mod p).) Then

from [45,Theorem 3] we have, for p = 20/+ 1 = a2 + b2 = e2 + 5/2, b z 0

(mod 5),

Mn a\ ( ]0f\ - [ lQf\ -2ea t A $20.4) [ f)=-\3f)=W\ {m0dpl

Let ß = 2(p~[)/l°. We show in the next two theorems that 8 representative

binomial coefficients are related to (wff) and (x°f) by powers of ß.

Theorem 20.1. Let p = 20 / + 1 =a2 + /32 = e2 + 5/2¿7ea prime with the signs of

a and e chosen so that a ee 1 (mod4), e ee a (mod5), if b = 0 (mod5) and e =\b\

(mod 5) if a ee 0 (mod 5). Then we have the following congruences modulo p:

2f\ -i ,i/^« „„ i uf2eaß(y) =(-$'2eß or (-1)



(2/)=(-l)2e^ or (-ly-pjj-,

according ai o ee 0 (mod 5) or a ee 0 (mod 5).

Proof. For the duration of this section all congruences are interpreted modulo

p = 20/ + 1 unless otherwise stated. By (3.11) and (2.2) we have

n-<)/(7),so the first congruence in Theorem 20.1 follows at once from (20.3) and (20.4).

Next we have

(20.6) <,,-,,,.,. = Œ| .(»)/13/\

6/)'

and the second congruence follows as above, noting that ß~3 = (~\)fß2 as (2/p) =

+ 1 =»/7 ee 1 (mod 8) ^/ ee 0 (mod2).

Similarly,

^^-r-IHÄ-lSrl/ivl-^vl/i?/)-yielding the third congruence in Theorem 20.1, and

completing the proof of Theorem 20.1.

Example. Let p = 241 = 142 + 5(3)2. Note that /= 12, 2(i,_l)/5 ee 1 (mod p),

2ea/\b\= 136 (mod241). In agreement with Theorem 20.1 we have

(SH-ISMïJ-ÎE)-» <-*»■Theorem 20.2. Le//7 = 20/ + 1 = a2 + o2 = e2 + 5/2 be aprime with the signs of

a and e chosen so that a ee 1 (mod 4), e = a (mod 5) if b = 0 (mod 5) ana" e =|è|

(mod 5) ;/ a ee 0 (mod 5). TVie« we /jaue Z/¡e following congruences modulo p:

4f\ =( n[2e/5]T„o /8/\=c ü/+l2f/5]i„o2_(-l)lze/D12e|S, "M =(-iri^J2e^2,

-(-ir'^'2^3, ^i



Proof. To prove the first congruence in Theorem 20.2 we need to show that

(20.9) (y)/(y)=(-l)/+[2e/5] or (_i)/+t2^llAl {modp)

according as o = 0 (mod 5), e ee a (mod 5), or a = 0 (mod 5),e=\b\ (mod 5).

The Jacobi sums J20(\, 1) and 4o(l, 3) are related by (see [31, Lemma 3])

(20.10) m40(1,1)=40(1,3),

so by Lemma 6 of [45] we must have

(20.11) (y)/(y)=« (mod/7).

Clearly (-l)'2*-/5) = +1 if e ee 1 (mod5) and (-l)'2^ = -1 if e es 4 (mod5).

We now use Lemma 4 of [31]. As e ee a (mod 5) if 6 ee 0 (mod 5), we have

/__C + 1 ifeEE 1 (mod5).

-1 ifeEE 4 (mod 5)

((-1)^ arising from a = ±(~\)f (mod5) in Lemma 4). Moreover, with ßi = gif =

a/\ b | (mod p), a = 0 (mod 5), we have

,_ \\b\/a ifeEE l(mod5),(-1) «—i

\a/\b\ ifeEE 4 (mod 5).

To prove the second congruence in Theorem 20.2 we first use (3.11) and (2.2) to

obtain

Next from Theorem 20.1, (20.3), and (20.4) we have

(20.13) (3/)/(TZ) ^H)^-'»/10)3.

Now we have

(20.14)8/! /!3/! 6/! 3/!7/! = / 8/\ / / 10/\

/!7/! 4/! 3/!3/! 10/! ~\4f}' \ 4f \

so

(20.15) W/M/U (2-"/10)4 ={_lY2(P-^v ; \ f!/ \ fl (.i/^-»/10)3

proving the second congruence in Theorem 20.2.

Next using (3.11), (2.1) and (2.2) we have

(20 16) (2(,-n/io)2 = iIiWi=^l^l^!_=(8/)/fv u o; K ' 2/! 12/! 2/! 10/! 6/! U/K I

10/

4//'



From Theorem 20.1, (20.3) and (20.4) we have

(20.17) (21/)/(y)-H)/(2<-'>/10)4

so

10/)

(20.18) (y/)/(4/)-(-l)/(2('-,,/10)2,

proving the third congruence in Theorem 20.2.

Finally, we have

ya,.») 4/!7/, 8/, 2/!6/, 10/! y 2/j/ \

so from (20.16) and (20.17) we deduce that

(20.20) (14^)/(8/)=(-l)/(2<'-I>/10)2,

completing the proof of Theorem 20.2.

Let (x, u, v, w) be a solution of

(20.21) 16/7 = x2 + 50m2 + 50u2 + 125tv2, x = l(mod5),

xw = v2 — 4uv — u2.

Example. Let p = 3121 so/ = 0 (mod2), 2(/,~1)/5 = +1, a =-39, 6 = 40 = 0

(mod 5), and e = -49 ee a ee 1 (mod 5). We have

™ (r)-(7)-(y)-(ï)-(î!0-(^■(y)-(y)-(^-(30—*-»» >•

Resolving the sign ambiguity in Cauchy's congruence (see (20.2)) involves showing

that C0/) and (30/) differ multiphcatively by 5(p_,)/4ee±1 (mod p). The con-

gruences in the following Theorem are related by a fourth root of unity, m, which

does not arise from any expression of the form (n(p~l)/m)', e = mn, t> 1. Thus

Muskat's and Whiteman's determination of u in Lemma 3 of [31] is an important

and valuable result. In our notation this determination takes the form

(20.24, „J«-')'"2"" «»-0(«odfl.

$-$'*U-ma/\b\ ¡fa SO (mod5>.



Theorem 20.3. Let p = 20/ + 1 = a2 + b2 = e2 + 5/2 be a prime with the signs of

a and e chosen so that a = 1 (mod 4), e ee a (mod 5) i/ft = 0 (mod 5) and e ee | b \

(mod 5) // a = 0 (mod 5). Then we have

(2/)/(4/)H^)/(^)^-'»f/)/(-)

-=<A"ivmi)/(v)-w(v)^MM"iHv-Vmv)/(v)-(30/(?)-(7)/(^-(y)/(?0--^"-

Proof. The first four congruences in Theorem 20.3 are an immediate consequence

of Theorems 20.1 and 20.2. The next two congruences follow from the first as

(20.25)

(y)"/0-(y)/(y)-o/O-(y)/(îi)-Next we have, using (2.2) and Theorems 20.1, 20.2

(20.26)

■(30/(30-^(ï)/(ÏT)-<-)"•(20.27)

(></p^m/(^H-'>i8;)/m-o/o-(y)/(?o--

(20.28)

(^'/O-(20/(^-0/0-completing the proof of Theorem 20.3.

Corollary. For every prime p = 20/ + 1 we have

(M-të)W(X)-(#*-*



As 2(/,~1)/5 is given explicitly by (4.13) we have, if (2/p)5 ¥= 1

(20.29) (y)=(-l)/+[2e/51

2e(w(125w2 - x2) + 2(xw + 5uv)(25w - x + 20m - 10u))

n>(125w2 - x2) + 2(xw + 5uv)(25w - x - 20m + lOu)

a EE 1 (mod4), e = a (mod 5) if b = 0 (mod 5) and e =\b\ (mod5) if a ee 0 (mod 5),

m EE 0 (mod 2), x + m — v = 0 (mod 4). Moreover, we have the following (using the

result of Emma Lehmer in formula (48) of [23]):

(20.30) ("/j-nr751

2e(w(125H<2 - x2) - 2(xw + 5uv)(25w + x + 10m + 20u))

w(\25w2 - x2) - 2(xw + 5mu)(25w + x - 10m - 20u)

if (2//7)5 ¥= 1 (« replace (x, m, u, h>) by (x, -u, m, -w) on the right-hand side of

(20.30)).Example. Let/7 = 41 so x = -9, u — 0, v — 3, w = -1, e = -6. Then we have

8/(If) =38 (mod 41)

and

/

8/) -r^l-W-Ui-lJ™ - 81) + 2(9)(-25 + 9 - 30))// v ' -1(125 - 81) + 2(9)(-25 + 9 + 30)

ee 12(-3 + 33)/(-3+ 6) = 38 (mod41).

Moreover, we have

('#) -(?)-» (-«»

and

H/\ =r y-,2/5] -12(-1(125 - 81)) - 2(9)(-25 -9 + 60)4// l ; -1(125 — 81) — 2(9)(-25 — 9 — 60)

ee 12(-3- 17)/(-3 + 11) ee 11 (mod 41).

Now (4f) = (2) = 28 and, replacing (x, u, v, w) by (x, v, -u, -w), we have

4/\ _ , y-i2/5]-12(125 - 81) + 2(-9)(25 + 9 + 60)

/ / [ ' (125 - 81) + 2(-9)(25 + 9-60)

= 12(3- 11)/(3 + 17) =28 (mod41).

Finally we have

f\ = (262) ee34 (mod 41)3/



and, replacing (x, m, v, w) by (x, -u, -v, w), we have

Í U/) = ( n[-i2/5)-12(-l(125 - 81) + 2(9)(-25 + 9 + 30))

\ 3/ / { ' -1(125 - 81) + 2(9)(-25 + 9-30)

= 12(-3 + 6)/(-3+ 33) = 34 (mod41).

The binomial coeffcients in the above corollary are of Cauchy-Whiteman type.

Moreover it is clear from the above congruences that

MWMv)are expressible in terms of the parameters in (20.21) rather than the parameter e in

p = e2 + 5/2.

Before proceeding to determine the binomial coefficients which may be given

explicitly in terms of the system (20.1 ) we note that the above theorems yield a large

number of congruences relating products and/or quotients of representative bi-

nomial coefficients (as in (20.1) or in Theorem 14.1) and the parameter e in

p = e2 + 5/2. We cite only a few.

<-» (7HH-"'(4/n)N-D'C/)^1/)^ (mod,),

<-> (7H'^<-<HS("1)/(2/)(121/) s4e2°r-4i>2 (mod^<

according as b = 0 (mod 5) or a = 0 (mod 5).

,20,3) [%*',)/["'f)^'^ N**>:

(20.34) (y)(4//)/(^)s2eor2£<!/|i.| (mod,,),


The congruence (20.32) may be obtained from Theorem 20.2 after noting that

— mViMMlViv)-l2f

f/(y) =uß4 (mod/7);



(20.31), (20.32), and (20.33) are clearly immediate consequences of Theorems 20.1

and 20.2. Also we note in view of Gauss's congruence given in Theorem 7.1 that we

have

(20.36) 2eaor-2e|o| (mod p)


Apart from ( ™J ) all 8 lower order binomial coefficients are given explicitly by the

congruences in §§8 and 13. We now prove that 6 representative binomial coefficients

of order 20 may be given explicitly in terms of the system (20.21).

Theorem 20.4. For each prime p = 20/+ 1 = a2 + ¿72 = e2 + 5/2, a = 1 (mod 4),

with(x, u, v, w) a solution of (20.21), set y +(x, u, v, w) (upper signs) and yl(x, m, v, w)

equal to

-[-x±w(x2 - 125w2)/4(xw + 5m«)]

if(2/p)s — 1, and to

;(125> x2) + 2(xw + 5uv)(25w - x + 20m - 10«)

;(125w2 - x2) + 2(xw + 5uv)(25w - x - 20m + 10«)

v(x7 I25w2

4(xw + 5m«)

if 2 is a quintic nonresidue of p. Then with e = a (mod 5) if b = 0 (mod 5), e =\b\

(mod 5) if a = 0 (mod 5), with a fixed primitive root g such that g5f = a/\ b | (mod p ),

and with u = 0 (mod 2), x + u — v = 0 (mod 4), we have the following congruences

(mod p).

3/= y+(x, u,v,w)

for the solution (x, -«, u, -w) o/(20.21), and

9f\ _3/

(-1)f+[2e/5]

y+(x,u, v,w)

or

(-l)f+l2e/5](a/\b\)y+(x,u,v,w)

according as b = 0 (mod 5), in which case the + sign holds and e = a (mod 5), or

a = 0 (mod 5), in which case the - sign holds ande =\b\ (mod 5).

Moreover, we have

(y)=(-i)V(x,M,«,w)

for the solution (x, -u, -«, w) of (20.21), and

/3/\ _ (9/\ _ ,V+l2e/S] , ,

\/] U/l Y.(x,M,«,w)



or

(-l)/+[2e/5](a/\b\)y_(x,u,v,w)

according as 6 = 0 (mod 5), in which case the + sign holds and e = a (mod 5), or

a = 0 (mod5), in which case the - sign holds and e=\b\ (mod5); (x, u, v, w) is

replaced by (x, -«, u, -w).

Proof. The first congruence follows directly from Theorem 20.2 noting that

[V\/[%f\- 7/!4/!4/!ll/! /H/\//H/\\f!/ \ 4// 3/!4/!8/!ll/! ~\3f}' \ 4f f '

The Cauchy-Whiteman type congruences

'7fl_. (9f\ __, /3/\_. {4Pf 3/

and/ 2/

(mod p ),

holding with the + sign if b = 0 (mod 5) and the - sign if a = 0 (mod 5), are proved

in [17]; the congruences stated in Theorem 20.4 for (J) and (3/) follow immediately

from Theorems 13.1 and 20.3.

Finally, the congruence for (9/) follows from Theorem 20.2, noting that (using

(2.2)and03 = (-l)//?2),

13/!/

= (-!)'

13/

4/)/

18/)/(11/

4/= (-l);/?3 (mod/7).

Example. Let/7 = 41 so(x, m, v,w) = (-9,0,3,-1) and

(j;) = ('64)-10 0-d41). (7/) = ('24)-9 0-41).

= ('68)ee-9 (mod 41).

We have

y+(x,-u, u,-w) = -

EE 10 (mod 41);

(-\)f+[2e/5](a/\b\)y+(x,u,v,w)

125 - 81 + 2(-9)(25 + 9-60)

125 - 81 + 2(-9)(25 + 9 + 60)9 + (81 - 125)

4(-9)

9" 2

Moreover,

-1(125 - 81) + 2(9)(-25 + 9-30)

-1(125 - 81) + 2(9)(-25 + 9 + 30)9 + (81 - 125)

4(-9)ee9 (mod41).

(9/)-(?)-»^». (y)-(í)=".

(2/) = (48) S-]5 <m<>d41>'



and we have

(-1) y_(x,-u,~v,w)

-1(125 - 81) + 2(9)(-25 + 9 + 30)

-(125-81) + 2(9)(-25 + 9

(-l)f+l2e/5](a/\b\)y_(x,-v,u,-w)

30)

(125 - 81) + 2(-9)(25 + 9-60)

(81 - 125)4(-9) '

(81 - 125)4(-9) '

30 (mod41),

EE 15 (mod 41).

and

/

12/

5/

and12/

5/

(125 - 81) + 2(-9)(25 + 9 + 60)

Having explicitly determined 16 of the 24 representative binomial coefficients of

order 20 in Theorems 20.1, 20.2, and 20.4 there remain

iHv)m]Mmi6,]In light of Theorem 20.3 it suffices to determine

5/\ ¡an ¡if)fl'XWW)

In the following theorem we determine these 4 binomial coefficients up to sign.

Clearly such complicated determinations are almost solely of theoretical interest.

However, we make no apology, as the same may be said of far simpler determina-

tions. Moreover, the proof of Theorem 20.5 yields some neat explicit determinations

for certain products and quotients of the remaining 8 representative binomial

coeffcients; these are enumerated in Theorem 20.6.

Theorem 20.5. Let p = 20/ + 1 = a2 + ¿72 = e2 + 5/2, define y+ (x, u, u, w) and

y_(x, u, v, w) as in Theorem 20.4, and for a fixed primitive root g with g5/ = a/\ b \

(mod p), choose the signs of a, e, x, u, and v so that e = a (mod 5) if b = 0 (mod 5),

e =| b | (mod 5) if a = 0 (mod 5), u = 0 (mod 2), and x + u - v = 0 (mod 4). 77ie

following congruences determine the binomial coefficients

./)•(!/)• (v)-and (v)modulop up to sign.

I fe )1/2[(-1) -y+(x,-u,-v,w)y_(x,u,v,w)j

I f e \ 1/2[(-1) T£-.y+(x,-u,-v,w)y_(x,u,v,w)j

5/

/

¡/¿7 = 0(mod5),

if a =0(mod5),

8/\ _\(4eay+(x,-v,u,-w)/y_(x,u,v,w)) ifb = 0 (mod 5),

3/' |(4e|o|Y+(x,-«,M,-w)/Y.(x,M,«,w))'/2 if a = 0 (mod 5),

7/

2/

/ ta y/¿((-1) -y+(x,-u,-v,w)y_(x,u,v,w)\ ifb = 0 (mod5),

/loi \ '/2l(-l) —y+(x,-u,-v,w)y_(x,u,v,w)\ if a ee 0 (mod5),



((-1) 4eay^(x, v,-u,-w)/y+ (x, u, «, w)) ifb = 0(mod5),

{(-\)f4e\b\yXx, v,-u,-w)/y+ (x, u, v,w)) if a = 0 (mod5).

Proof. We have

12/)

5/j

(-) (5;)/(7H:í)/"0/5/

5/W ( 9/\ _ / 10/\ / / 10/\ _ efV U/H/J/ \5f)=-a °r W\

according as b = 0 (mod 5) or a = 0 (mod 5) in view of (20.3), (20.4) and Theorem

7.1. Moreover,

(20.38) (5/)(^) = (9/)(4/) =(-Vfy+(x,-u,-v,w)ySx,u,v,w)

in view of Theorem 20.4 (the mapping (x, u, v, w) -* (x, -u, -«, w) for either

Y+ (x, m, «, w) or y_(x, u, u, w) has the effect of multiplying by ß3 = (2(^~1)/10)3).

Combining (20.37) and (20.38) we have the first congruence in Theorem 20.5.

Next, we have

(20.39) (|J)/(?/)-(J)/(!Jf)-n(*^...-)/t.(x......)in view of Theorems 8.1 and 20.4. Also, using (2.2), we have

(20.40)

l*f)3//1

^/(30"v«.(^/(^)-(-./(3f)/(-i/(^)

3/) ( 5/^ E

according as b = 0 (mod 5) or a = 0 (mod 5) in view of (20.3), (20.4), and Theorem

7.1. Combining (20.37) and (20.38) we have the second congruence in Theorem 20.5.

Now we have

(20.41)

,vW'/O-ß)/(305/\ /{V\ _/10/\ /[\0f\ _e

2/

\2f)/\2fJ=[ 3/)/ \ 5fJ~a "' \b\

according as b = 0 (mod 5) or a = 0 (mod 5) by (20.3), (20.4), and Theorem 7.1.

Also,

(2°-42> ( 2/) ( 2/) = (3/)(24/) =H>V(*. —' »h-(x,u,v,w)

in view of Theorem 20.4 and (13.8) (noting that /?8 = (-\)fß3). Combining (20.41)

and (20.42) we have the third congruence in Theorem 20.5.



Finally, combining (20.39) and (20.40) and noting that (x, u, v, w) -> (x, -u, m, -w)

has the effect of multiplying by ß4, (x, u, v, w) -» (x, «, -u, -w) of multiplying by ß,

and \/ß4 = (-\)fß, we have the last congruence in Theorem 20.5.

The following theorem is an immediate consequence of (20.37), (20.40), (20.41)

and similarly derived congruences together with Theorem 20.3.

Theorem 20.6. Let p = 20/ + 1 = <z2 + o2 = e2 + 5/2 with a and e chosen as in

Theorem 20.4 and with g a fixed primitive root such that gif = a/\b\ (mod p). Then

we have

or

5/

,2/

)/(vH!V(l(S)/ß)-(y)/ßH

../ß)-(?)/ß)-Tiiaccording as b = 0 (mod 5) or a = 0 (mod 5). Moreover,

or

ï)(30-<-^(y)(ïo-'-according as b — 0 (mod 5) or a = 0 (mod 5),

/(ïl-^/iî)-«-^a/jí?

according as e = 1 (mod 5) or e ee 4 (mod 5).

Example. Let/7 = 641 so (x, m, u, w) = (16,4,12,-4), 2(;>"1)/5 ee 1 (mod5)(«x

ee 0 (mod 2)), a = 25, b — 4, e = -6, and / = 32. We have, in agreement with

Theorem 20.5, the following congruences modulo 641 :

(5}f=(lSîsi3 and f(434^191^13'

8/)

3/j= i256) =443 and (4)(-6)(4)(434)/191 = 443,

(2/) =(2644) S564 and -4(434)(l9l)/-6 ee 564,

l5f)2(l6s)2-10 and (4K-6)(4)(191)/434 = 70,



for it is easily checked that

. . -16 -4(256 - 2000) 199 ... , ._,.Y+(x, m, «, w) = —-1-7-+-=——=434 (mod 641),,+ K ' 2 8(-64 + 240) 126 v ;'

Y_(x, m, u, w) = 191 (mod 641).

Moreover, we have as e ee 4 (mod 5),

(MZ)-w-{sDA%)'^'T'ï (-«»•

(a/)/(ír)-^-(^)/(?)-l-7=é (-«')■

( 3/) ( S/7) =(468)(46°) ^ 4e|o|EE - ( y) ( y7) ee(330)(280) (mod641),

(sYKs/) ^(^(^^^«'^(yjly) =(330)(468) (mod641).

21. e = 24. For p = 24/ + 1 there are 33 representative binomial coefficients of

order 24 and 15 lower order representatives. An astonishing 43 of these 48 binomial

coefficients may be related to at least one other by what we henceforth call a

Cauchy-Whiteman type congruence, that is, a congruence relating two representative

binomial coefficients of the type.

for all p — ef + 1. (The name derives from the facts that Cauchy proved

10/\ _ MO'/ / - - \3fj (mod^)

for all p = 20/ + 1 and Whiteman showed how to remove the sign ambiguity in this

congruence).

We begin this section by using the Davenport-Hasse relation in the form given by

Yamamoto (3.11), together with (2.1), (2.2), and (2.6), to prove all Cauchy-White-

man type congruences for e = mn — 24 in which representatives are related by a

term of the form (nip~l)/m)'= ±\,t> 1.

For the rest of this section, all congruences are understood to be taken (mod p =

24/+ 1) unless otherwise stated.

Theorem 21.1. The following congruences hold for all prime p = 24/+ 1 — a2 + b2,

a ee 1 (mod 4); a = 1 if a = 0 (mod 3) and a = 2 ifb = 0 (mod 3).

y)-(wy)-<-«i3f



11/

/

(6/

13/

= (-1)

9/) =(-\)f+h/\

2/1 l } \4//' \ f

4f\ -2/

12/\

/ / "\ 5/

/+!,/*( 14/

4/

i9/W10/U-nW13/4/

).(-i)-(2y),(2y),(-i)-(^.

(-iyw\=(_,)fiw5/ 2/

H/\ (8/

5/ ' 2/

= (-1) f+b/4 12/

3/ )-'"«O ($-<-«>■(

14/

6/

10/

4/

(^)^(-l)—(12/ i/12/2/,S<-'»;*S/V/

10/

2/

16/)

8/

Proof. The theorem follows directly from the following 19 congruences, noting

that 2 and 3 are quadratic residues of every prime p = 24/+ 1, 2(/,~1)/4 = (-1)*/4

(mod p) (Gauss) and 3<p~"/4 = l or -1 (mod p), according as b = 0 (mod 3) or

a = 0 (mod 3).

(21.1)

(21.2)

(21.3)

(21.4)

(21.5)

(21.6)

(21.7)

fl 15/)

Congruence

6/)=( l)/-/4i12/|

3/i * U 3/

9/

3/

6/

2/

12/

2/

10/

2/

8/

2/

(-1)

= (-!)"

= (-!)"

16/

8/

14/

6/

/+Í./4 12/

6/

10/

4/

12/

6/

Reason

Theorem 11.2

Theorem 11.2

Corollary 4.1.3

Theorem 15.1

Corollary 4.2.2

Corollary 4.4.1.

Theorem 11.2

Next, using (3.11), (2.1), and (2.2) we have

yp-n/8-3/!8/!16/i , -3/!

(3(P-0/8

/ ! 9/ ! 17/ ! / ! 9/ ! 17/ !

= H)/3/!23/!14/! ,v ; 9/! 17/! 14/! * u

9/ ! 8/ ! 16/ !3/! 11/! 19/! <-»'( 30/

10/

/

19/

9/

'/(?

13/

3/ /(14/\

5/i


496 R. H. HUDSON AND K S. WILLIAMS

It follows that

Moreover,

12/

/

.-^"-(-./(SOffl/ISHI 13/! 5/!9/! 3/!7/! 10/!

1 ' 3/! 10/! 14/! 10/! /!9/!

_5/!7/!6/! _[lf\/[\\f\/!ll/!6/! \f)/ \ 5fj-

/ I 12/\ = 12/! 5/!7/!6/! = / lf\ / I ll/\

/ \ 5fj /!11/! 12/!6/! ~\f)' \ 5/ }

lf\ 5/!6/!4/! = /5/\ / I Uf\" /!6/! ll/!4/! " \ f)/ \ 4fY

Combining these we have

v\ _/ii/(21.8) \fl-\5f

[7HH

Next using (3.11) and (2.2) we have

2r.-n/»-2/112/Ill/! _ ( 12/\ / / 13/\

■ /! 13/! 11/! \ f )' \ 2fj'

0/125=iopnpip yiun/iun{¿ > ~ 5/!17/!7/! -( l) \5fj/\ If Y

Taking g = 12, h = 1, k = 2 in (2.6), and using (2.2), gives

(7)(7HyHK'>iya-Making use of (21.9) we obtain

1 = 2(/'-,>/2 =(2</'-1»/12)(2(/'_1)/12)5

giving

2/\ _ / 14/1(21.11)

In establishing the remaining congruences we will use (21.8)—(21.11) without

specifically citing them.



Taking g = 14, h = 1, k = 12 in (2.6) we have

<-)ir)/(^)^-'»f/)/(r)H,7)/(^)— *•from which follows

(21.12) l5/)"(-1)\2//-10/\ _, ,,// 13/

Appealing to (3.11), we get

9(p-D/3 _ i7(p-\)/\i\* = 8/! 12/! 8/!

v ; 4/!16/!8/!

30/(?)-(30/(£rmoreover,

/lO/Wl3/\ /M3/Wl2/\ _ 10/! 4/19/! /! 11/! 13/!V/ }\2f)' \ 4fJ\ f j f\9f\ 13/! 12/! 2/! 11/!

_ 4/! 10/! 12/! _ /l2/\ /l\0f\

2/!12/!8/! \ 4/// 2//'V10/\ _, ,,ft/4/ 13/

and it follows that

(2..13) | ■;)-(-!) l4//.

Since

10/\ / / 13/\ _ 10/!4/!3/! _ 10/!4/!14/!V/ // \4fj /!13/!3/! /!13/!14/! '

we obtain, using (2.2),

(21-14) i4/)=(-Dfc/4/l3// / v ' \ 3/

H/\ _/ ^/+h/A(W

(2U5) (/),(-,) l4/

Next, using the previously established congruence for 2<p~1)/l2,

M/\ //7/\ /12/\ //4/\_ 12/!5/!5/!/mnv2/// \2// \ 2/// \2/, 7/!10/!5/!

/(';/) =(-»'-'12/l /



From (15.7) we have

2<p-')/3 12/W4/

2/rU/so

(21.16) (Z)^-l^ÍZ)

Since

4/\ /¡If] _ 4/!5/!9/! _ 4/!5/!3/!V2/// \2fj 2/!7/!9/!

we have at once that

p.."» (Z)^-'y"'(Z(21.18) (25/r)-(-')W4(3/

Now, appealing to (2.1), (2.2), and (3.11) we have

(60>-17/4)3 =3_ 18/ ! 4/ ! 8/ ! 12/ ! 16/ ! 20/ !

3/! 7/! 11/! 15/! 19/! 23/!

= (-1) /+l| 18/!12/!5/!/!13/!6/!

\ 3/! 7/! 11/! 13/! 6/!

i-1)'! \8/) ( u/f)i2/! 5/! /! 6/!/6/! i8/!

wA/yimThus, using Theorem 11.2, we obtain

This completes the proof of Theorem 21.1.

The number of Cauchy-Whiteman type congruences in Theorem 21.1 exceeds that

of all such congruences for all lower order cases (e < 24). Perhaps, even more

surprisingly, it does not include all the Cauchy-Whiteman type congruences for

e = 24. In contrast to the lower order cases (and anything we can find elsewhere in

the literature), there are Cauchy-Whiteman type congruences for e = mn — 24 for

which the ± 1 relating

rf\ I r'f\and ,, modulo p

sf) \sf)

is not an expression of the form (n(p~X)/m)', t~>\. We have, instead, the following

theorem which, in conjunction with Theorem 21.1, gives all Cauchy-Whiteman type

congruences for e = 24.



Theorem 21.2. The following congruences hold for all primes p = 24/+ 1 = a2 +

b2 = x2 + 3y2 = m2 + 6u2, a = u=\ (mod 4), x = 1 (mod 3):

r/H-irf/), r/H-oi-).7/\=(_1)/+W11/)> (4'U(_i)/+W8/

TH-^li. (9/H-'>ll10/\ =(_1)/+«/2( 15/j> / 13/\ =(_ir/2/ 13/)

/ / V ' \ 7/ j ' \ 2/ / v ' \ 5/ / ■

Proof. From Berndt [3, p. 3.23] we have

/24(l,4) = (-l)/+ü/22<"-'»/12/24(8,8),

so that, appealing again to Theorem 5.1 we have that for a prime ideal divisor -n of P

in Q(e2,"/24),

(yj-i-l)^^-"/'2)5!1^) (mod.)

^|5/j=(_1)/+,/4(2(,-l)/6)jl6/j (mod77)

(as 2(^"/4 = (-\)h/4 and (-l)fc/4+l,/2 = (-l)'/4 follows from [3,pp. 317,3.25]).

Next Theorems 6.2 and 9.2 give

p/C«) .*-.<-.,in view of 4.5 (and (2<^_I>/3)2 = 2(p-],/6). It follows at once that

(5;H-»-('8y) <-„.Moreover, we have using

ÏM2from Theorem 21.1,

ivmymwczrgiving all congruences for which ± 1 = (-\)f+y/4.

Next, using Theorem 21.1,

/VSl-(ï)/[$W[®-™


500 R. H HUDSON AND K. S. WILLIAMS

and, using

from Theorem 21.1, we have

mzi-w/w-M/mwAWgiving all congruences for which ± 1 = (-1)'/2.

Finally, using (2.2) and

(30-të)from Theorem 21.1, we have

=<-»f/)/(r>(-./-.completing the proof of Theorem 21.2.

Theorem 21.3. Let p = 24/+ 1 = a2 + b2 = u2 + 6u2 with a = u = 1 (mod4).

77rew we Äaue, according as b = 0 (mod 3), or a ee 0 (mod 3), i/ze following con-

gruences:

( 7H v'H-^2" or (~i)/+'2m (mod/,)-

Proof. From Berndt and Evans [4, pp. 374,377] we have

/24(12,23) = m - ivfi for m EE 1 (mod 4) ;

moreover, applying Theorem 5.1 and using (2.2), we have

/24(12,23) = -(iy)EE(-ir'(iy) (mod.).

As /24(1' 12) — 0 (mod w) we obtain, as before,

EE(-l)/+12U(mod/7) for mee 1 (mod 4).

From Berndt [4, Theorem 3.18] we have u = 1 (mod 4) iff a = 0 (mod 3) and u = -1

(mod 4) iff b EE 0 (mod 3), completing the proof of Theorem 21.3 in view of Theorem

21.1.

Using previously established congruences from §§5,6,7,9,11,15 and Theorems

21.1-21.3, we now prove the following theorems.

12/

/



Theorem 21.4. Let p = 24/+ 1 = a2 + b2 = x2 + 3y2, a = 1 (mod4), x ee 1

(mod 3). Let a = 1 // a = 0 if a = 0 (mod 3), ß = 1 // a or b = 2 (mod 3), a = 2 //

o ee 0 (mod 3), ß — 2 if a or b ee 1 (mod 3). We have the following congruences

modulo p.

>»/12/\ _/l2/)

6/Hr6/4(^H-iyl2/ 2a,

(Tl-^lf/l^-^fv)(-ir 14/)

6/j2a or 2/3

according as b = 0 (mod 3) or a = 0 (mod 3),

(-1)£a/4/

2/(-1)

4 + a( 7/M _

2/.

(-1)/4 + a/H/\ =,_n«/l4/\

/ / l" M 4/ )

2 a

2ax — 6a>>

x + 3 k<

2ax + 6ay<

x — 3^

2a

2ax + 6ay-

x — 3y

2ax — 6ay

x + 3y

if y = Q(mod3),

if y = 1 (mod 3),

if y = 2 (mod 3),

if y =0(mod3),

if y = 1 (mod 3),

if y = 2 (mod 3).

Proof. The first three congruences follow immediately from Theorems 11.1, 15.1,

and 7.1, respectively. The congruences for

8/\ /4/\ l\4f\2//'U//' and \4/

follow from Theorem 15.1 and the remaining congruences then follow from Theo-

rems 21.1 and 21.2.

Theorem 21.5. Let p = 24/+ 1 = c2 + 2a"2, c ee 1 (mod 4). Then we have

(-ir(^) -h/I"/) ^)f+b/\7f)=2< (^dp).

Proof. This is immediate from Theorem 11.2.

Theorem 21.6. Let p = 24/+ 1 = a2 + b2 = x2 + 3/, 4p = A2 + 21B2, a = 1

(mod 4), A ee x ee 1 (mod 3). Lei ß = 1 // a or b ee 2 (mod 3), ß = 2 if a or b = \

(mod 3). 77ie7? we /zaue the following congruences modulo p.

\2f\4/

= 2x,

(-D»(26^2, , 2i,V»-(-D''(!^)s2 x or 2ax/b




10/

2/

16/

8/

(-1)/+.V/4 5/

2x //> = 0(mod3),

-x — 3y if y ee 1 (mod 3),

-x + 3y if y ee 2 (mod 3),

8/l4/j " \ 4/

2x ;/>> = 0 (mod 3),

-x + 3y if y = 1 (mod 3),

-x — 3y if y = 2 (mod 3).

Proof. The first congruence in Theorem 21.6 follows from Corollary 4.1.1 and

Theorem 9.2 (also from (15.4) and Theorem 15.1). The rest of the congruences in this

theorem follow easily from Theorems 6.1, 6.2, 15.1, and 21.2.

Theorem 21.7. Let p = 24/+ 1 = a2 + b2 + 3/ = m2 + 6«2, u = 1

(mod 4), x = 1 (mod 3). Let a = 1 // a = 0 (mod 3), ß= I if a or b = 2 (mod 3),

a = 2 if b = 0 (mod 3), ß — 2 if a or b = 1 (mod 3). We have the following

congruences modulo p.

12/

/

'^H-D-2,

/ ,\/+o/2 + j)| 6/\ _. ,,,/2 + a + W 13/\ _ - .,

// \6/)= °r W/


(_ir+a|io/j^(1) f+h/4 + ai 13/ _(_1)/+-v/4 + ai 13/

2flK> \5f

(-1)/4 + a| 2/

/

2m //>; = 0(mod3).

(2xm - 6yu)/ (x + 3y) if y = 1 (mod 3),

(2xm + 6yu)/ (x — 3y) ify = 2 (mod 3),

+«/8/:/_j\/+.v/4 + al oy 1 ^/_jn/+6/4

2m ify = 0(mod3),

(2xw + 6yu)/ (x — 3y) ify = 1 (mod 3),

(2xm — 6vm)/(x + 3^) ify = 2 (mod 3).



Proof. Theorem 21.3 constitutes the first two congruences in this theorem.

To prove the last six congruences in Theorem 21.7 we use (3.11) and (2.2) to

obtain

2,„-n/.,-2/!12/!ll/! _/12/\ //13/

■ /!13/!11/! \ f )/ \2f

_ 2/! 12/!/! //2/W/l2/\

/! 13/! 11/! l M /// I / /'

The last six congruences follow immediately from the above congruence and

Theorems 21.1-21.3.

To prove the third and fourth congruences in Theorem 21.7 we note that earlier

(see (21.3) and the congruence following (21.7)) we showed that

9/\ //12/\

3/!6/! 12/! /!9/! 10/! 5/!

6/)/('52/)-

The result now follows from Theorems 21.1 and 21.3.

Remark. The first author has shown (unpublished) that the criteria of Hudson

and Williams [16] for 3 to be an eighth power (mod p) is derivable from the above

theorems (in a more general form). This derivation is neater than the one given in

[16] requiring cyclotomy. A complete determination of the Jacobi sums of order 40

would, in our opinion, undoubtedly lead to a criteria for determining when s(p~,)/s

ee +1, -1, b/a, or -b/a (mod p) in terms of the parameters a, b, l, m in p = a2 +

b2 — I2 + 10m2. This would be of interest if it is new and as simple as the classical

criteria.

Finally, the remaining binomial coefficients of order 24 can be determined up to

sign as in §§18-20. We cite just one example, as the details are easily obtained.

Theorem 21.8. Let p = 24f + \ = a2 + b2 = c2 + 2d2 = x2 + 3y2 = m2 + 6«2,

a = c = m = 1 (mod 4), x ee 1 (mod 3). 77ie« we have

where the binomial coefficients on the right-hand side of the congruence are given in

Theorems 21.4-21.7.



Proof. The theorem follows immediately from the easily established congruences

and

(7)/(>(7)/P-Remark. As before each of the binomial coefficients, which we are only able to

determine up to sign, is completely determined if the sign ambiguity can be removed

for any one of these.

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Department of Mathematics and Statistics, University of South Carolina, Columbia, South

Carolina 29208

Department of Mathematics and Statistics, Carleton University, Ottawa, Ontario K1S 5B6,

Canada


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