abc conjecture implies infinitely many non-x …...abc conjecture implies in nitely many...
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ABC Conjecture Implies Infinitely Manynon-X-Fibonacci-Wieferich Primes – A
Linear Case of Dynamical Systems
Jun-Wen Wayne Peng
Department of Mathematics, University of Rochester
3/21/2016
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Motivation
The Fibonacci sequence {Fn}∞n=0, an ancient and beautiful integersequence, is the following:
0, 1, 1, 2, 3, 5, 8, 13, . . . .
This sequence is originally defined by the following recursiverelation, with initial values F0 = 0 and F1 = 1,
Fn = Fn−1 + Fn−2 ∀n ≥ 2.
Let α > 0 and α be the roots of x2 − x − 1 = 0. Then, the closedform of Fn is given by
Fn =αn − αn
√5
.
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Motivation
It will be very interesting to consider this sequence modulo m forsome integer m. Let m be 11, and then the sequence {Fnmod 11} is
0, 1, 1, 2, 3, 5, 8, 2, 10, 1, 0, 1, 1, . . . .
Let m = 7, the sequence is
0, 1, 1, 2, 3, 5, 1, 6, 0, 6, 6, 5, 4, 2, 6, 1, 0, 1, 1, . . . .
We should note that, in the both cases, the sequences repeat 0, 1,and so form a period. In fact, we can define a function
π(m) := the period of the Fibonacci sequence modulo m.
In 1960, D.D. Wall gave a paper in AMM(The AmericanMathematical Monthly), and studied the properties of the functionπ in that amusing paper.
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Motivation
Conjecture (wall’s conjecture)
π(p) 6= π(p2) ∀p.
Definition
We call a prime p Fibonacci-Wieferich prime if
π(p2) = π(p).
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Motivation
We can treat more generally. Given a degree d linearly recurrencesequence X = {xm} with closed form
xm = b1am1 + b2a
m2 + · · ·+ bda
md
for ai , bi ∈ K \ {0} where K is a number field, we can define, forsome prime ideal p of K ,
πX (pe) := period of {xm mod pe}
where e is an integer, and X is not degenerated modulo p, i.e.ai , bi 6≡ 0 mod p, and we then ask whether
πX (p) = πX (p2)
or not. We will call these primes p X -Fibonacci-Wieferichprimes.
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ABC Conjecture Implies Infinitely Manynon-X-Fibonacci-Wieferich Primes – A
Linear Case of Dynamical Systems
Jun-Wen Wayne Peng
Department of Mathematics, University of Rochester
3/21/2016
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Wieferich Prime
A Wieferich Primes is a prime satisfying the following equation
2p−1 ≡ 1 mod p2.
It has been conjectured that there are infinity Wieferich Primesalthough only two have been found so far. These two Wieferichprimes are
1093 and 3511.
In stead of proving Wieferich primes infinitely many, we can maybeshow that the non-Wieferich primes are finite.Unfortunately, in 1988, Silverman proved that there are infinitelymany non-Wieferich primes assuming ABC conjecture, so it is stilluncertain that the number of Wieferich primes is finite or infinite.
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Wieferich Prime
What is the relation between Wieferich primes and Wall’sconjecture?
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Connection
We first give a degree d linearly recurrence sequence X , and thengive a prime ideal p of K where X is not degenerated modulo p.Then, X mod pe is period of length π if and only if
b1aπ1 + · · ·+ bda
πd ≡ x0 mod p
b1aπ+11 + · · ·+ bda
π+1d ≡ x1 mod p
...
b1aπ+d−11 + · · ·+ bda
π+d−1d ≡ xd−1 mod p
holds.
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Connection
If we consider each aπi to be a variable Xi , then the above systemof equations can be expressed as
b1 b2 · · · bdb1a1 b2a2 · · · bdad
......
. . ....
b1ad−11 b2a
d−12 · · · bda
d−1d
X1
X2...Xd
≡x0x1...xd
mod pe
The matrix is a generalized Vandermonde matrix, so the thesystem has a unique solution which we should note that[1, 1, · · · , 1]T is a solution.
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Connection
Therefore, we have aπ1aπ2...aπd
≡
11...1
mod pe ,
and it implies that
π = πX (pe) = lcm{ordpe (a1), ordpe (a2), . . . , ordpe (ad)}
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Observation
πX (p) = lcm{ordp(a1), ordp(a2), . . . , ordp(ad)}
πX (p) | pf − 1
ordp(ai ) 6= ordp2(ai )⇒ ordp2(ai ) = p ordp(ai )
If X = {a1, a2} and a1a2 = ±1 (multiplicative dependent), then anX -Fibonacci-Wieferich prime is just an a1-Wieferich prime.Otherwise, we only get
p is an ai -Wieferich prime⇒ p is an X -Fibonacci-Wieferich prime.
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An Easy but Useful Lemma
Definition
Let SQFP(I ) be the square-free part of an ideal I ⊂ K , i.e.
SQFP(I ) =∏p‖N
p.
Lemma (P.)
If p is a prime dividing SQFP(ani − 1) for some n, then p is not anX -Fibonacci-Wieferich prime.
So, if we can show that SQFP(αn − 1) goes to infinity as n goes toinfinity, we prove that there are infinitely manyx-Fibonacci-Wieferich primes.
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A Brief History
Mathematicians were interested in Wieferich primes, because itwas proved that if p is not a Wieferich prime and is the first caseof Fermat’s last theorem, then Fermat’s last theorem will hold onthis prime p. Two Chinese mathematicians, Zhihong Sun andZhiwei Sun (they are brothers), show that the Fibonacci-Wieferichprimes have the same property in 1992, so we sometimes callWieferich-Fibonacci primes Wall-Sun-Sun primes, which arereferred to the potential counterexamples of Fermat’s last theorem.However, Fermat’s last theorem was proved a year later in 1993.
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abc conjecture
We need several definitions for stating the ABC conjecture.
Definition (Places of K)
Let K be a number field, and MK be all places of K . A finite placev ∈ MK , correspondent to a prime ideal P, is defined as
‖x‖v = |NK/Q(P)|v(x) ∀x ∈ K .
An infinite place v ∈ MK , correspondent to an embeddingσ : K → C, is defined as
‖x‖v = |σ(x)|e ∀x ∈ K
where e = 1 if σ is totally real, otherwise, we let e = 2.
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abc conjecture
Definition (Logarithm Height)
A logarithm height function h on PN(K ) is, for all[x0 : . . . : xN ] ∈ PN(K ),
h([x0 : . . . : xN ]) =1
[K : Q]
∑v∈MK
log max{‖x0‖v , ‖x1‖v , . . . , ‖xN‖v}
Definition (set of primes of a point)
A set of prime ideals of a point p = [x0 : . . . : xN ] ∈ PN(K ) is
I ([x0 : . . . : xN ]) = {P ⊂ OK | vP(xi ) 6= P(xj) for some i , j}
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ABC conjecture
Definition (radical of a point)
Radical of a point p ∈ PN(K ) is
rad(p) =∏P∈I (p)
|N(P)|.
where we define N := logNK/Q/[K : Q].
Conjecture (ABC)
Let p ∈ P2(K ) be a point on the hyperplane X + Y + Z = 0. Forany given ε > 0, we can find a constant Cε such that
h(p) < (1 + ε) rad(p) + Cε(�ε).
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Our Result
We call a field K ABC-field if ABC conjecture is true over K .
Theorem (p.)
There are infinitely many non-Wieferich-Fibonacci primes assumingK to be an ABC-field.
For demonstrate our ideal, we can simply consider ABC conjectureover Q.
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Skech of Proof
Let cn = an + bn where a, b ∈ Z with gcd(a, b) = 1.Claim: |Un| = |SQFP(cn)| → ∞ as n→∞ (Write cn = UnVn)Suppose NOT! Say |Un| < C for some C for all n.Apply ABC, ∀ε > 0, we have
h(|an|, |bn|, |cn|)�ε (1 + ε) rad(an, bn, cn). (1)
We havelog |Un|+ log |Vn| ≤ h(an, bn, cn) (2)
and
rad(an, bn, cn) < log |Un|+1
2log |Vn|+ log |ab|. (3)
(1)+(2)+(3)⇒
1
1 + ε(log |Un|+ log |Vn|)�a,b,ε log |Un|+
1
2log |Vn|
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Sketch of Proof
Then, since we assume |Un| is bounded
1
1 + εlog |Vn| �ε,a,b,C
1
2log |Vn|,
a contradiction when ε < 1.
Remark
This theorem tells that vn cannot too big (,i.e. has a upper bound).
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Another Result of Applying n-term ABC Conjecture
If we consider the canonical trace Tr : K → Q given by
Tr(k) =∑
σ:K ↪→C
σ(k) ∀k ∈ K ,
then
Theorem (P.)
Let α be an algebraic integer with degree greater than 1. If then-term ABC conjecture is true over the splitting field of α, thenTr(αm)s are not perfect power for all m large enough.
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Lower Bound
We define
Wα(B) := {p | N(p) ≤ logB, and p is an α-Fibonacci-Wieferich prime.},
and we are going to show that there exists some constant Cα suchthat
|Wα(B)| ≥ Cα logB
for all B greater enough.
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Idea
We hope p‖(γm − 1) where m is the first integer such thatp | (γm − 1), and it is well-known that
γm − 1 =∏d |m
Φd(γ).
This goal can be achieve by showing that p‖Φm(γ), i.e.p|SQFPφm(γ) =: Un, and showing p - Φd(γ) for any other d < m.This means γ is a root of Φm(x) mod p, but it is not a root ofΦd(x) mod p. Then, we note that if we suppose p and m arecoprime and γ is not a ”bed” element modulo p, then xm − 1mod p is separable, i.e. when γ is a root of some factor Φd(x)with d |m, it won’t be a root of others. Therefore, our first goalcan be make by assuming that p| SQFP(Φm(γ)), γ is not a pole orzero divisor modulo p and p and n are coprime.
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Idea
We should also expect that the prime that we give in the latestslide is not a γ-Fibonacci-Wieferich prime These give me the firstlemma.
Lemma
Let OK be the ring of integer of K , write (γ) = IJ−1 withI + J = OK . If we assume (n)IJ + p = OK and p ⊇ SQFP(Φn(γ)),then we have
mp = n and γNK/Q(p)−1 6≡ 1 mod p2.
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Idea
Now, if we let
N(SQFP(Φn(γ))) > log(n)N(IJ), (4)
then we automatically get a prime p satisfying
p + nIJ = OK and p | SQFP(Φn(γ)).
We want to count how many n do we have for having (4), so wedefine
{n | N(SQFP(Φn(γ))) > log(n)N(IJ)}.
We then find that we have a one-to-one map
{n | N(SQFP(Φn(γ))) > log(n)N(IJ)} ↪→ Mγ
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Idea
Then, it remains to figure out the constant B ′ for which
|Wγ(B)| ≥ |{n ≤ B ′ | N(SQFP(Φn(γ))) > log(n)N(IJ)}|,
and it gives the second lemma.
Lemma
|Wγ(B)| ≥ |{n ≤ (logB − log 2)/h(γ) | N(Un) > log(n)N(IJ)}|.
For simplify the notation, we define the following notations:
(γn − 1) = unvnw−1n where uv + w = OK and u = SQFP(unvn)
(Φn(γ)) = UnVnW−1n where UV +W = OK and U = SQFP(UnVn)
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Idea
|N(Unvn)| ≥ |N(UnVn)| ≥ |N(Φn(γ))|
N(vn) N(Un) ≈ ϕ(n)∫∂D1(0)
log |γ − z |dz
Bounded above by ABC What we want ≥ cϕ(n)
⇒ We get a lower bound, which is
cϕ(n)− nεh(γ)− Cε,γ ,
for the |N(Un)|
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Idea
We then let this lower bound greater than log(n)N(IJ), whichequivalently is
cϕ(n)− nεh(γ)− Cε,γ − log(n)N(IJ) ≥ 0 (5)
A theorem from analytic number theory comes to our rescue
Lemma
|{n < Y | ϕ(n) ≥ δn}| ≥ (6
π2− δ)Y + O(logY )
Thus, for arbitrary δ > 0, if we choose ε = cδ/(2h(γ)) and letϕ(n) ≥ δn, then
cϕ(n)− nεh(γ) ≥ cδn − 1
2cδn =
1
2cδn > 0
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Main Theorem
Theorem
For any non zero algebraic number γ ∈ X \ ∂D1(0) where ∂D1(0)is the complex unit circle with center at origin, there is a constantCγ such that we have
|Wγ(B)| ≥ Cγ logB ∀B � 0
assuming K is an abc-field.
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Heuristic Result: Algebraic version
Why do we conjecture infinitely many Wieferich primes? Even ifwe only find two Wieferich primes.
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Let p be a prime integer. Considering
2p ≡ 2 + kp mod p2 for k ∈ {0, 1, . . . , p − 1},
we suppose k is uniformly distributed on the set {0, 1, . . . , p − 1},which means that the possibility of getting k = 0 is 1/p. Thus, weexpect to have ∑
p≤X
1
p≈ log logX
many Wieferich primes under X .
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Heuristic Result: Algebraic version
We use the similar idea to compute the number of{2, 3}-Fibonacci-Wieferich primes and should have∑
p
1
p2≈ 0.4522474200410654985065.
More generally, if we assume a1, . . . , ad are multiplicativelyindependent, then the number of X -Fibonacci-Wieferich primes is∑
p
1
pd.
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Dynamical System
Arithmetic dynamics is refer to the study of the number-theoreticproperties of the iteration of polynomials or rational functions. Letf : X → X be an endomorphism. We denote the following:
1 f n =
n times︷ ︸︸ ︷f ◦ f ◦ · · · ◦ f
2 A point p ∈ X is periodic if f n(p) = p for some n > 1.
3 The orbit of p is the set
Of (p) := {p, f (p), f 2(p), f 3(p), . . .}
4 Let f ∈ OK [x ], and let ⊆ OK be an ideal of K . We denotep := p mod I for p ∈ OK . Then, we define
Of (p; I ) := {p, f (p), f 2(p), f 3(p), . . .}
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Heuristic: Geometric Version
Thus, via the terminology of arithmetic dynamics, a linearrecurrence sequence {xn}, which has a recurrence relation
xm+d+1 = c1xm+1 + c2xm+2 + · · · cdxm+d
with initial values x1, x2, . . . , xd , can be expressed by given thefunction f : Rd → Rd with
f : (xm+1, xm+2, . . . , xm+d) 7→ (xm+2, xm+3, . . . , xm+d+1)
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Heuristic: Geometric Version
More generally, we can consider an algebraic variety X over theconstant field K , an endormorphism f of X and a point on X . Weshould ask:
Do we have infinitely many primes p such that
|Of (p, p)| = |Of (p, p2)|? (6)
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Heuristic: Geometric Version
We propose the following conjecture.
Conjecture
1 We have infinitely many primes p such that (6) holds ifdimOf (p) = 1;
2 we have finitely many primes p such that (6) holds ifdimOf (p) > 1.
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Data: 2n − 1
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Data: 2n + 3n
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Data: the Fibonacci Sequence
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Thank you!