MST Topics inHistory of Mathematics

Euclid’s Elements,the Works of Archimedes,

and the Nine Chapters of Mathematical Art

Paul Yiu

Department of MathematicsFlorida Atlantic University

Chapters 11–13

Summer 2017

June 30, 2017

Contents

11 Euclid’s Elements, Book VII 30111.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30111.2 The euclidean algorithm . . . . . . . . . . . . . . . . . . . . . . . 30111.3 Euclid I.4-10 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30311.4 Numbers in proportion . . . . . . . . . . . . . . . . . . . . . . . . 30411.5 Commutative law of multiplication . . . . . . . . . . . . . . . . . 30511.6 Euclid VII.17-20 . . . . . . . . . . . . . . . . . . . . . . . . . . . 30711.7 Prime and composite numbers . . . . . . . . . . . . . . . . . . . . 30711.8 Least common measure . . . . . . . . . . . . . . . . . . . . . . . 30911.9 Appendix: The fundamental theorem of arithmetic . . . . .. . . . 309

12 Euclid’s Elements, Book VIII 311

13 Euclid’s Elements, Book IX 31513.1 Euclid IX.1-19 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31513.2 Continued proportions . . . . . . . . . . . . . . . . . . . . . . . . 31513.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31613.4 Infinitude of prime numbers . . . . . . . . . . . . . . . . . . . . . 31713.5 Even and odd numbers . . . . . . . . . . . . . . . . . . . . . . . . 31813.6 Summation of geometric progression . . . . . . . . . . . . . . . .32013.7 Perfect numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . 321

13.7.1 Euler’s proof . . . . . . . . . . . . . . . . . . . . . . . . . 322

Chapter 11

Euclid’s Elements, Book VII

11.1 Definitions

Definitions (VII.1). A unit is that by virtue of which each of the things that existis calledone.(VII.2). A number is a multitude composed of units.(VII.3). A number is apart of a number, the less of the greater, when itmeasuresthe greater;(VII.4). But parts when it does not measure it.(VII.5). The greater number is amultiple of the less when it is measured by theless.

Remarks.(1) The unit (one) is not a number, but is a part of every number.(2) A number measures another (greater) number if the greater number is a

multitude composed of the less.(3) Definition 4 does not simply mean that one number does not measure an-

other. As the proof ofEuclid VII.4 reveals, to say thatA is parts ofB means thatA is a sum of numbers (or units) each of which is a part ofB.

11.2 The euclidean algorithm

The first two propositions determine if two given numbers areprime to one another,and if not, to find their greatest common measure.

Euclid (VII.1). Two unequal numbers being set out, and the less being continuallysubtracted in turn from the greater, if the number which is left never measures theone before it until a unit is left, the original numbers are prime to one another.

Euclid’s proof . For the less of two unequal numbers AB, CD being continuallysubtracted from the greater,let the number which is left never measure the one before it until a unit is left.

302 Euclid’s Elements, Book VII

B AF H

D CG

E

Figure 11.1: Euclid VII.1

Claim: AB, CD are prime to one another, i.e., a unit alone measures AB, CD.For, if AB, CD are not prime to one another, some number will measure them.Let a number measure them them, and let it be E;let CD, measuring BF , leave FA less than itself,let AF , measuring DG, leave GC less than itself,and let GC, measuring FH, leave a unit HA.

Since E measures CD, and CD measures BF , E also measures BF .But it also measure the whole BA,therefore it also measures the remainder AF .But AF measures DG, therefore E also measures DG.But it also measure the whole DC,therefore it will also measure the remainder CG.But CG measures FH,therefore E also measures FH.But it also measures the whole FA;therefore it will also measure the remainder, the unit AH, though it is a number,which is impossible.Therefore no number will measure the numbers AB, CD;therefore AB, CD are prime to one another.

Euclid (VII.2)). Given two numbers not prime to one another, to find theirgreatestcommon measure.(Porism). From this it is manifest that, if a number measures two numbers, it willalso measure their greatest common measure.

In modern notations, and allowing the notion of negative integers, we proveEuclid VII.2.

Given two numbersa and b, assuminga > b, we construct two sequencesa2, a3, . . . , an andq1, q2, . . . , qn−1 as follows.(i) a1 = a anda2 = b.(ii) For each numberi, let qi be the greatest number of timesai+1 can be subtractedfrom ai, i.e.,qiai+1 ≤ ai and(qi + 1)ai+1 > ai. Setai+2 = ai − qiai+1. Note thatai+2 < ai.Therefore the sequences terminate whenan+1 = 0 for somen. This means thatan

11.3 Euclid I.4-10 303

measuresan−1.If an measuresan−i and an−i−1, then it also measuresan−i−2 sincean−i−2 =an−i + qn−ian−i−1.Sincean measuresan andan−1, it also measuresan−2, an−3, . . . ,a2 = b anda1 = a.an is a common measure ofa andb.On the other hand, every common measure ofa andb measuresa3, . . . ,an.Therefore,an is the greatest common measure ofa andb.

Corollary. The greatest common measure of two numbers is the difference of mul-titudes of the numbers.

Notation:(A,B) = greatest common measure ofA andB.

Euclid (VII.3). Given three numbers not prime to one another, to find the greatestcommon measure.

The greatest common measure ofA, B, C is ((A,B), C). This extends to morethan three numbers.

11.3 Euclid I.4-10

Euclid (VII.4). Any number is either a part or parts of any number, the less of thegreater.

For numbersA less thanB, eitherA measuresB, or B is a multitude of thegreatest common measure ofA andB.

Euclid.(VII.5). If a number be part of a number, and another be the same part ofanother,then the sum will also be the same part of the sum that the one is of the one.

A =1

nB, C =

1

nD =⇒ A+ C =

1

n(B +D).

(VII.6). If a number be parts of a number, and another the same parts ofanother,the sum will also be the same parts of the sum that the one is of the one.

A =m

nB, C =

m

nD =⇒ A+ C =

m

n(B +D).

(VII.7). If a number be that part of a number, which a number subtractedis of anumber subtracted, the remainder will also the same part of the remainder that thewhole is of the whole.

A =1

nB, C =

1

nD =⇒ A− C =

1

n(B −D).

304 Euclid’s Elements, Book VII

(VII.8). If a number be the same parts of a number that a number subtracted is of anumber subtracted, the remainder will also be the same parts of the remainder thatthe whole is of the whole.

A =m

nB, C =

m

nD =⇒ A− C =

m

n(B −D).

(VII.9). If a number be a part of a number, and another be the same part of another,alternately also, whatever part of parts the first is of the third, the same part, or thesame parts, will the second also be of the fourth.

A =1

nB, C =

1

nD,A =

p

qC =⇒ B =

p

qD.

(VII.10). If a number be parts of a number, and another be the same partsof an-other, alternately also, whatever parts of part the first is ofthe third, the same parts,or the same part will the second also be of the fourth.

A =m

nB, C =

m

nD,A =

p

qC =⇒ B =

p

qD.

11.4 Numbers in proportion

Definition (VII.20). Numbers areproportional when the first is the same multiple,or the same part, or the same parts, of the second that the third is of the fourth.

Euclid (VII.11). If, as whole is to whole, so is a number subtracted to a numbersubtracted number, the remainder will also be to the remainder as whole to whole.

Assumea > a′, b > b′, anda : b = a′ : b′. Thena− a′ : b− b′ = a : b.

Euclid (VII.12). If there be as many numbers as we please in proportion, then asone of the antecedents is to one of the consequents, so are allthe antecedents to allthe consequents.

a : a′ = b : b′ = · · · =⇒ (a+ b+ · · · ) : (a′ + b′ + · · · ).

Proof:Let A, B, C, D be as many numbers as we please in proportion, so that, as A is toB, so is C to D;I say that as A is to B, so are A, C to B, D.For since, as A is to B, so is C to D,whatever part or parts A is of B,the same part of parts is C of D also (Def. VII.20)Therefore, also the sum of A, C is the same part or the same parts of the sum ofB, D that A is of B.Therefore, as A is to B, so are A, C to B, D.

11.5 Commutative law of multiplication 305

Euclid (VII.13). If four numbers be proportional, they will also be proportionalalternately.

a : a′ = b : b′ =⇒ a : b = a′ : b′.

Euclid (VII.14). If there be as many numbers as we please, and others equal tothem in multitude, which taken two and two are in the same ratio, they will also bein the same ratioex aequali.

If a : b = a′ : b′ andb : c = b′ : c′, thenex aequali, a : c = a′ : c′.

11.5 Commutative law of multiplication

Definition (15). A number is said tomultiply a number when that which is multi-plied is added to itself as many times as there are units in theother, and thus somenumber is produced.

3 multiplies5 : 5 + 5 + 5,5 multiplies3 : 3 + 3 + 3 + 3 + 3.

We readA×B asA multipliesB (orB multiplied byA), and it means the sumof A copies ofB:

A×B = B + B + · · ·+ B (A copies).

Euclid VII.16 is the commutative law of multiplication of numbers:

A×B = B × A.

Euclid (VII.15). If a unit measures any number, and another number measures anyother number the same number of times, then alternately also, the unit will measurethe third number the same number of times that the second measures the fourth.

Let A be the unit andB = pA. Consider a third numberC measuring a fourthoneD the same number of times,D = pC. Then, if C = qA, it follows thatD = qB.Proof:For let the unit A measure any number BC, and letanother number D measure any other number EF the same number of times;I say that, alternately also, the unit A measures the number Dthe same number of times that BC measure EF .

For since the unit A measures the number BCthe same number of times that D measures EF ,

306 Euclid’s Elements, Book VII

therefore, as many units as there are in BC,so many numbers equal to D are there in EF also.

Let BC be divided into the units in it, BG, GH, HC, ...and EF into the numbers EK, KL, LF , ... equal to D. Thus,the multitude of BG, GH, HC, ... will be equal tothe multitude of EK, KL, LF , ...,and the numbers EK, KL, LF , ... are also equal to one another, whilethe multitude of the units BG, GH, HC, ... is equal tothe multitude of the numbers EK, KL, LF , ....

Therefore, as the unit BG is to the number EK,so will the unit GH be to the number KL,and the unit HC to the number LF .

Therefore also,as one of the antecedents is to one of the consequents,so will all the antecedents be to all the consequents. (VII.12)Therefore, as the unit BG is to the number EK,so is BC to EF .

But the unit BG is equal to the unit A,and the number EK to the number D.Therefore, as the unit A is to the number D,so is BC to EF .Therefore, the unit A measures the number Dthe same number of times that BC measures EF .

Euclid (VII.16). If two numbers multiplied by one another make certain numbers,then the numbers so produced equal one another.

Proof: Let A, B be two numbers,and let A by multiplying B make C,and B by multiplying A makes D;I say that C is equal to D.For since A by multiplying B has made C,therefore B measures C according to the units in A.But the unit E also measures the number A according to the units in it;therefore the unit E measures A the same number of times that B measures C.Therefore, alternately, the unit E measures the number B the same number oftimes that A measures C. (VII.15)

Again, since B by multiplying A has made D,therefore A measures D according to the units in B.But the unit E also measures B according to the units in it;

11.6 Euclid VII.17-20 307

therefore the unit E measures the number B the same number of times that Ameasures D.But the unit E measured the number B the same number of times that A measuresC;therefore A measures each of the numbers C, D the same number of times.Therefore C is equal to D.

11.6 Euclid VII.17-20

Euclid. (VII.17). If a number by multiplying two numbers make certain numbers,the numbers so produced will have the same ratio as the numbersmultiplied.

A×B : A× C = B : C.

(VII.18). If two numbers by multiplying any number make certain numbers, thenumbers so produced will have the same ratio as the numbers multipliers.

A× C : B × C = A : B.

Euclid (VII.19). If four numbers be proportional, the number produced from thefirst and fourth will be equal to the number produced from the second and third;and, if the number produced from the first and fourth be equal to that producedfrom the second and third, then the four numbers will be proportional.

A : B = C : D ⇔ A×D = B × C.

Euclid (VII.20). The least numbers of those which have the same ratio with themmeasure those which have the same ratio the same number of times; the greater thegreater; and the less the less.

If A : B = C : D andA, B are the least possible, thenC = nA =⇒ D = nB.

11.7 Prime and composite numbers

Definitions (VII.11). A prime number is that which is measured by a unit alone.(VII.12). Numbersrelatively prime are those which are measured by a unit aloneas a common measure.(VII.13). A composite numberis that which is measured by some number.(VII.14). Numbersrelatively composite are those which are measured by somenumber as a common measure.

Euclid (VII.21). Numbers prime to one another are the least of those which havethe same ratio with them.

308 Euclid’s Elements, Book VII

If A andB are prime to one another, then the ratioA : B is in its lowest terms.

Euclid (VII.22). The least numbers of those which have the same ratio with themare relatively prime.

If A : B = C : D andA, B are least possible, thenA andB are prime to oneanother.

Definitions (VII.16). And, when two numbers having multiplied one another makesome number, the number so produced be calledplane, and its sides are the num-bers which have multiplied one another.(VII.17). And, when three numbers having multiplied one another make some num-ber, the number so produced be calledsolid, and its sides are the numbers whichhave multiplied one another.(VII.18). A squarenumber is equal multiplied by equal, or a number which is con-tained by two equal numbers.(VII.19). And acube is equal multiplied by equal and again by equal, or a numberwhich is contained by three equal numbers.

Notation: A|B meansA measuresB.

Euclid (VII.23-28). LetA andB be relatively prime numbers.(VII.23) If C|A, thenC andB are relatively prime.(VII.24). If C andB are also relatively prime, then so areAC andB.(VII.25). A2 andB are relatively prime.(VII.26). A2 andB2 are relatively prime.(VII.27). A3 andB3 are relatively prime; so arean andbn.(VII.28). A+ B andB are relatively prime, and conversely.

Note that from VII.24, VII.25 follows by puttingC = A;from VII.25, VII.26 follows by puttingB andA for A andB in VII.25;

Proof of VII.27: gcd(A2, B2) = 1 ⇒ gcd(A3, B2) = 1 ⇒ gcd(A3, B3) = 1.

Euclid. (VII.29). Any prime number is prime to any number which it does not mea-sure.(VII.30). If two numbers by multiplying one another make some number, and someprime number measures the product, it will also measure one ofthe original num-bers.

If P is prime andP |AB, thenP |A or P |B.

Euclid. (VII.31). Any composite number is measured by some prime number.(VII.32). Any number is either prime or is measured by some prime number.

11.8 Least common measure 309

11.8 Least common measure

Euclid (VII.33). Given as many numbers as we please, to find the least of thosewhich have the same ratio with them.

GivenA, B, C, . . . , to find the least numbersA′, B′, C ′, . . . such that

A′ : B′ : C ′ : · · · = A : B : C : · · · .

If D = (A,B,C, . . . ), thenA = DA′, B = DB′, C = DC ′ etc.

Euclid (VII.34). To find the least number which two given numbers measure.

This is the number of times the greatest common measure ofA andB measuresAB. We shall writeLCM(A,B).

Euclid (VII.35). If two numbers measure any number, the least number measuredby them will also measure the same.

A|C andB|C =⇒ LCM(A,B)|C.

Euclid (VII.36). To find the least number which three given numbers measure.

LCM(A,B,C) = LCM(LCM(A,B), C).

Euclid (VII.37). If a number be measured by any number, the number which ismeasured will have a part called by the same name as the measuring number.(VII.38). If a number has any part whatever, it will be measured by a number calledby the same name as the part.

If A is measured byB, say,A = nB, thenA has a partC corresponding to themeasuring numbern.

Euclid (VII.39). To find the number which is the least that will have given parts.

11.9 Appendix: The fundamental theorem of arith-metic

The Fundamental Theorem of Arithmetic was first formulated and proved in Gauss’Dis-quistiones Arithmeticae, Section II, Theorem 16.

Theorem 11.1(Gauss,DA, Art.16). A composite number can be resolved into prime factorsin only one way.

310 Euclid’s Elements, Book VII

Demonstration. It is clear from elementary considerations that any composite number canbe resolved into prime factors, but it is often wrongly taken for granted that this cannot bedone in several different ways. Let us suppose that a composite number A = aαbβcγ , etc.,with a, b, c, etc. unequal prime numbers, can be resolved in still another way into primefactors. First it is clear that in this second system of factors there cannot appear any otherprimes excepta, b, c, etc., since no other prime can divideA which is composed of theseprimes. Similarly in this second system of factors none of the prime numbersa, b, c, etc.can be missing, otherwise it would not divide A (preceding article).1 And so these tworesolutions into factors can differ only in that in one of them some prime numberappearsmore often than in the other. Let such a prime be p, which appears in one resolution m

times and in the othern times, and letm > n. Now remove from each system the factorp,n times. As a resultp will remain in one systemm − n times and will be missing entirelyfrom the other. That is, we have two resolutions into factors of the numberA/pn. One ofthem does not contain the factorp, the other contains itm−n times, contradicting what wehave just shown.

1Section II, Congruences of the first order, ofDisquisitiones Arithmeticaebegins with the fol-lowing. 13. Theorem. The product of two positive numbers each of which is smaller than a givenprime number cannot be divided by this prime number. 14. If neither a nor b can be divided by aprime number, the productab cannot be divided byp. Gauss remarked that “Euclid had alreadyproved this theorem in hisElements(Book VII.No. 32). However we did not wish to omit it becausemany modern authors have offered up feeble arguments in place of proof or have neglected the the-orem completely, and because by this very simple case we can more easily understand the nature ofthe method which will be used later for solving much more difficult problems”. 15. If none of thenumbersa, b, c, d etc. can be divided by a primep neither can their productabcd etc.

Chapter 12

Euclid’s Elements, Book VIII

Euclid (VIII.1) . If there be as many numbers as we please in continued proportion,and the extremes of them are relatively prime, the numbers are the least of thosewhich have the same ratio with them.(VIII. 2). To find numbers in continued proportion, as many as may be prescribed,and the least that are in a given ratio.Porism).If three numbers in continued proportion are the least of those which havethe same ratio with them, then the extremes are squares, and, if four numbers, cubes.

Euclid (VIII.3) . If as many numbers as we please in continued proportion are theleast of those which have the same ratio with them, then the extremes of them arerelatively prime.(VIII.4)). Given as many ratios as we please in least numbers, to find numbers incontinued proportion which are the least in the given ratios.(VIII. 5). Plane numbers have to one another the ratio compounded of theratios oftheir sides.(VIII. 6). If there are as many numbers as we please in continued proportion, andthe first does not measure the second, then neither does any other measure anyother.(VIII. 7). If there are as many numbers as we please in continued proportion, andthe first measures the last, then it also measures the second.(VIII. 8). If between two numbers there fall numbers in continued proportion withthem, then, however many numbers fall between them in continued proportion, somany also fall in continued proportion between the numbers which have the sameratios with the original numbers.(VIII. 9). If two numbers are relatively prime, and numbers fall between them incontinued proportion, then, however many numbers fall between them in continuedproportion, so many also fall between each of them and a unit incontinued propor-tion.(VIII. 10). If numbers fall between two numbers and a unit in continued propor-tion, then however many numbers fall between each of them and a unit in continued

312 Euclid’s Elements, Book VIII

proportion, so many also fall between the numbers themselvesin continued propor-tion.(VIII. 11). Between two square numbers there is one mean proportional number,and the square has to the square the duplicate ratio of that which the side has to theside.(VIII. 12). Between two cubic numbers there are two mean proportional numbers,and the cube has to the cube the triplicate ratio of that which the side has to theside.(VIII. 13). If there are as many numbers as we please in continued proportion, andeach multiplied by itself makes some number, then the products are proportional;and, if the original numbers multiplied by the products makecertain numbers, thenthe latter are also proportional.(VIII. 14). If a square measures a square, then the side also measures theside; and,if the side measures the side, then the square also measures the square.(VIII. 15). If a cubic number measures a cubic number, then the side also measuresthe side; and, if the side measures the side, then the cube also measures the cube.(VIII. 16). If a square does not measure a square, then neither does the side mea-sure the side; and, if the side does not measure the side, thenneither does the squaremeasure the square.(VIII. 17). If a cubic number does not measure a cubic number, then neither doesthe side measure the side; and, if the side does not measure the side, then neitherdoes the cube measure the cube.(VIII. 18). Between two similar plane numbers there is one mean proportional num-ber, and the plane number has to the plane number the ratio duplicate of that whichthe corresponding side has to the corresponding side.(VIII. 19). Between two similar solid numbers there fall two mean proportionalnumbers, and the solid number has to the solid number the ratio triplicate of thatwhich the corresponding side has to the corresponding side.(VIII. 20). If one mean proportional number falls between two numbers, then thenumbers are similar plane numbers.(VIII. 21). If two mean proportional numbers fall between two numbers, thenthenumbers are similar solid numbers.(VIII. 22). If three numbers are in continued proportion, and the first issquare, thenthe third is also square.(VIII. 23). If four numbers are in continued proportion, and the first is acube, thenthe fourth is also a cube.(VIII. 24). If two numbers have to one another the ratio which a square number hasto a square number, and the first is square, then the second is also a square.(VIII. 25). If two numbers have to one another the ratio which a cubic numberhasto a cubic number, and the first is a cube, then the second is also a cube.(VIII. 26). Similar plane numbers have to one another the ratio which a squarenumber has to a square number.

313

(VIII. 27). Similar solid numbers have to one another the ratio which a cubic num-ber has to a cubic number.

Chapter 13

Euclid’s Elements, Book IX

13.1 Euclid IX.1-19

Definition (VII.21). Similar plane and solid numbers are those which have theirsides proportional.

Euclid (IX.1). If two similar plane numbers by multiplying one another make somenumber, the product will be square.

Let AB be a plane number. A similar plane number is one of the formA′B′

whereA′ : B′ = A : B. Using the commutative law (VII.16) and the associativelaw (tacitly assumed in Euclid).(AB)(A′B′) = · · · (AB′)(AB′), a square number.

Euclid (IX.2). If two numbers by multiplying one another make a square number,they are similar plane numbers.(IX.3). If a cube number by multiplying itself make some number, thenthe productwill be cube.(IX.4). If a cube number by multiplying a cube number makes some number, theproduct will be cube.(IX.5). If a cube number by multiplying any number make a cube number,the mul-tiplied number will also be cube.(IX.6). If a number multiplying itself make a cube number, it will itself also be cube.

Euclid (IX.7). If a composite number by multiplying any number make some num-ber, the product will be solid.

13.2 Continued proportions

Euclid (IX.8). If as many numbers as we please beginning from a unit are incon-tinued proportion , the third from the unit will be square, as will also those whichsuccessively leave out one; the fourth will be cube, as will also all those which leave

316 Euclid’s Elements, Book IX

out two, and the seventh will be at once cube and square, as will also those whichleave out five.

If a1 = 1, a2 = a, anda1, a2, a3, . . . , is a geometric progression, thenan =an−1, anda3, a5 etc are squares,a4, a7 etc are cubes, anda7, a13 etc are all ”cubeand squares”.

Euclid (IX.9). If as many numbers as we please beginning from a unit be in con-tinued proportion, and the number after the unit is square, all the rest will also besquare. And, if the number after the unit be cube, all the restwill also be cube.

If a is a square (cube) number, so is everyan.

(IX.10). If as many numbers as we please beginning from a unit be in continuedproportion, and the number after the unit be not square, thenneither is any othersquare except the third from the unit and all those which leaveout one. And, if thenumber after the unit be not cube, neither will any other cube except the fourth fromthe unit and all those which leave out two.

Euclid (IX.11). If as many numbers as we please beginning from a unit be in contin-ued proportion, the less measures the greater according to some one of the numberswhich have place among the proportional numbers.(Porism). And it is manifest that, Whatever place the measuring number has, reck-oned from the unit, the same place also has the number according to which it mea-sures, reckoned from the number measured, in the direction of the number before it.

(IX.12). If as many numbers as we please beginning from a unit be in continuedproportion, by however many prime numbers the last is measured, the next to theunit will also be measured by the same.

If any term of a geometric progression is measured by a prime number,then the secondterm is measured by the same prime number.

(IX.13). If as many numbers as we please beginning from a unit be in continued pro-portion, and the number after the unit be prime, the greatestwill not be measuredby any except those which have a place among the proportional numbers.

13.3

Euclid (IX.14). If a number be the least that is measured by prime numbers, thenit is not measured by any other prime number except those originally measuring it.

A prime number is not measured by any other prime number.If p1, p2 andq are distinct prime numbers, andp1p2 is not measured byq.

13.4 Infinitude of prime numbers 317

If p1, p2, . . . ,pn are distinct prime numbers, thenp1p2 · · · pn is the least numbermeasured by them.

If q is a prime number different fromp1, . . . ,pn, then the product is not measuredby q.

Euclid (IX.15). If three numbers in continued proportion be the least of those whichhave the same ratio with them, any two whatever added together will be prime tothe remaining number.

If a, b, c are relatively prime and in geometric progression, then each of thenumber is relatively prime to the sum of the remaining two.

Euclid (IX.16). If two numbers be prime to one another, the second will not be toany other number as the first is to the second.

If a andb are relatively prime, there is no geometric progressiona, b, c (in whichc is a (whole) number).

Euclid (IX.17). If there be as many numbers as we please in continued proportion,and the extremes of them be prime to one another, the last will not be to any othernumber as the first is to the second.

If a1, a2, . . . , an is a geometric progression in whicha1 andan are relativelyprime, thena1, a2, an, a do not form a geometric progression for any numbera.

Euclid (IX.18). Given two numbers, to investigate whether it is possible to findathird proportional to them.(IX.19). Given three numbers, to investigate when it is possible to finda fourthproportional to them.

Two given numbersa andb can be extended into a geometric progressiona, b,c if and only if a measuresb2.

13.4 Infinitude of prime numbers

Euclid (IX.20). Prime numbers are more than any assigned multitude of primenumbers.

Proof. LetA, B, C be the assigned prime numbers;I say that there are more prime numbers thanA, B, C.

For let the least number measured byA, B, C be taken, [VII. 36]and let it beDE;let the unitDF be added toDE.

318 Euclid’s Elements, Book IX

ThenEF is either prime or not.

First, let it be prime;then the prime numbersA, B, C, EF have been foundwhich are more thanA, B, C.

Next, letEF not be prime;therefore it is measured by some prime number. [VII. 31]Let it be measured by the prime numberG.I say thatG is not the same with any of the numbersA, B, C.For, if possible, let it be so.NowA, B, C measureDE;thereforeG also will measureDE.But it also measureEF .ThereforeG, being a number, will measure the remainder, the unitDF :which is absurd.

ThereforeG is not the same with any one of the numbersA, B, C.And by hypothesis it is prime.Therefore the prime numbersA, B, C, G have been foundwhich are more than the assigned multitude ofA, B, C.

13.5 Even and odd numbers

Definitions (IX.6). An even numberis that which is divisible into two equal parts.(IX.7). An odd number is that which is not divisible into two equal parts, or thatwhich differs by a unit from an even number.

How do we decide if a (large) number is even or odd? How do we know if alarge pile of chips is even or not? We count by twos (pairs). The number is evenif the counting can be finished without leftover (remainder). It can be divided intotwo equal parts by choosing one from one pair. If the pile cannot be finished withpairs, there must be one left over, and the rest is an even number.

Euclid (IX.21). If as many even numbers as we please be added together, the wholeis even.

Euclid’s proof: Consider a bunch of even numbers. Each of these has a halfpart. The sum of these half parts, one for each even number, isa half part of thesum. Therefore the sum is even.

Alternative proof: Each even number can be counted by pairs.Given a numberof even numbers, we keep on counting the numbers by pairs and finish with all of

13.5 Even and odd numbers 319

them. The sum is therefore even, its half part is the sum of a half part from each (ofthe given) even number.

Euclid (IX.22). If as many odd numbers as we please be added together, and theirmultitude be even, the whole will be even.

If we count by pairs, each odd number has a unit leftover. There are an even number ofunit remainders, which can be counted by pairs.

(IX.23). If as many odd numbers as we please be added together, and theirmultitudebe odd, the whole will also be even.

If we count by pairs, each odd number has a unit leftover. There are an odd numberof unit remainders. Counting by pairs, we have at the end one unit leftover. Therefore, thesum is odd.

Euclid (IX.24). If from an even number an even number is subtracted, the remain-der is even.

An even number consists of an exact number of pairs.

(IX.25). If from an even number an odd number is subtracted, the remainder is odd.Start with an even number dividing into pairs. Separate one of the pairs into two units.

Subtracting an odd number means removing one unit and a number of pairs. In the end wehave a number of pairs and a unit. The difference is odd.

(IX.26). If from an odd number an odd number is subtracted, the remainder is even.An odd number is a number of pairs, together with a unit. Subtracting an oddnumber

from an odd number amounts to removing the unit and a number of pairs. In the end wehave a number of pairs. The difference is even.

(IX.27). If from an odd number an even number is subtracted, the remainder is odd.

Euclid (IX.28). If an odd number by multiplying an even number make some num-ber, the product will be even.

Euclid’s proof: For let the odd numberA by multiplying the even numberBmakeC;I say thatC is even.For sinceA by multiplyingB has madeC,thereforeC is made up of as many number equal toB as there are units inA.(Def.VII.15)AndB is even;thereforeC is made up of even numbers.But, if as many even numbers as we please be added together, thewhole is even.(X.21)

320 Euclid’s Elements, Book IX

ThereforeC is even. Q.E.D.

Euclid’s proof did not actually assumeA odd. It works for any numberA.

Euclid (IX.29). If an odd number by multiplying an odd number make some num-ber, the product will be odd.

Euclid (IX.30). If an odd number measure an even number, it also measures halfof it.(IX.31). If an odd number be prime to any number, it will also be prime to thedouble of it.

Definitions (IX.8). An even-times even numberis that which is measured by aneven number according to an even number.(IX.9). An even-times odd numberis that which is measured by an even numberaccording to an odd number.(IX.10). An odd-times odd number is that which is measured by an odd numberaccording to an odd number.

Euclid (IX.32). Each of the numbers which are continually doubled beginning froma dyad iseven-times even only.

These are powers of2.

Euclid (IX.33). If a number have its half odd, then it is even-times odd only.(IX.34). If a number neither be one of those which is continually doubled from adyad, nor have its half odd, it is both even-times even and even-times odd.

13.6 Summation of geometric progression

Euclid (IX.35). If as many numbers as we please be in continued proportion, andthere be subtracted, from the second and the last, numbers equal to the first, then,as the excess of the second is to the first, so will the excess of the last be to all thosebefore it.

If a1, a2, . . . ,an, an+1 is a geometric progression, then

(an+1 − a1) : (a1 + a2 + · · ·+ an) = (a2 − a1) : a1.

Equivalently,

a1 + a2 + · · ·+ an =a1

a2 − a1(an+1 − a1)

13.7 Perfect numbers 321

Proof.

a2

a1=

a3

a2= · · · =

an

an−1

=an+1

an

=⇒a2 − a1

a1=

a3 − a2

a2= · · · =

an − an−1

an−1

=an+1 − an

an

=(a2 − a1) + (a3 − a2) + · · ·+ (an − an−1) + (an+1 − an)

a1 + a2 + · · ·+ an−1 + an

=an+1 − a1

a1 + a2 + · · ·+ an−1 + an.

Corollary. 1 + 2 + 22 + · · ·+ 2n−1 = 2n − 1.

13.7 Perfect numbers

Definition (VII.22). A perfect number is that which is equal to the sum its ownparts.

For example,

6 = 1 + 2 + 3,

28 = 1 + 2 + 4 + 7 + 14

are perfect numbers. At the end of the number theory books, Euclid gives thefollowing construction of (even) perfect numbers.

Euclid (IX.36). If as many numbers as we please beginning from a unit are set outcontinuously in double proportion until the sum of all becomes prime, and if thesum multiplied into the last makes some number, then the product is perfect.

In modern notations, supposeMn = 1+2+22 + · · ·+2n−1 is a prime number,then2n−1 ·Mn is a perfect number. Note that

1 + 2 + 22 + · · ·+ 2n−1 = 2n − 1.

n 2n − 1 perfect number2n−1(2n − 1)5 31 16 · 31 = 4967 127 64 · 127 = 812813 8191 4096 · 8191 = 33550336

Euclid IX.36 has historically led to several important questions.

322 Euclid’s Elements, Book IX

(1) The quest of prime numbers of the formMn := 2n − 1. It is easy to see thatn must be a prime ifMn is a prime. The converse needs not be true. To date (May,2014), there are only48 known Mersenne primes.1

(2) The perfect number constructed in IX.36 are even numbers. Euler (173?)has shown that indeed this construction gives all even perfect numbers.

(3) The existence of anodd perfect number is still an open problem.(i) Benjamin Peirce (1832, Harvard professor 1833-1880) proved that if an

odd perfect number exists, it must contain at least4 distinct prime divisors.(ii) Current record: P. Nielsen, Odd perfect numbers have at least9 distinct

prime factors, 2007.(iii) There is no odd perfect number< 10300.

13.7.1 Euler’s proof

Euler made use of the sum-of-divisors functionσ(P ), which has the multiplicativeproperty:σ(PQ) = σ(P )σ(Q) for relatively primeintegersP andQ. In terms ofthe functionσ, a numbern is perfect if and only ifσ(n) = 2n.

Let P be anevenperfect number. WriteP = 2n−1 · Q for n − 1 ≥ 1 andQconsisting only of odd divisors. Since2n−1 andQ are relatively prime,

σ(P ) = 2P

=⇒ σ(2n−1)σ(Q) = 2n ·Q

=⇒ (2n − 1)σ(Q) = 2n ·Q.

It follows that

σ(Q) =2nQ

2n − 1= Q+

Q

2n − 1.

Note that this last expression is an integer. This means that2n − 1 is a divisor ofQ, so is Q

2n−1. Sinceσ(Q) is the sum ofall divisors ofQ. This equation shows

that Q

2n−1must be1, andQ has no more divisors apart fromQ and1. From this we

conclude thatQ = 2n − 1 and is a prime number (since its only divisors areQ and1). The even perfect number isP = 2n−1(2n − 1) constructed in Euclid IX.36.

1The latest oneM57885161 has17425170 digits, and was discovered in 2013. The correspondingperfect number has34850339 digits.

13.7 Perfect numbers 323

n Number of digits Year Discoverer

2 1 Ancient3 1 Ancient5 2 Ancient7 3 Ancient13 4 145617 6 1588 P. A. Cataldi19 6 1588 P. A. Cataldi31 10 1750 L. Euler61 19 1883 I. M. Pervushin89 27 1911 R. E. Powers107 33 1913 E. Fauquembergue127 39 1876 E. Lucas521 157 1952 R. M. Robinson607 183 1952 R. M. Robinson1279 386 1952 R. M. Robinson2203 664 1952 R. M. Robinson2281 687 1952 R. M. Robinson3217 969 1957 H. Riesel4253 1281 1961 A. Hurwitz4423 1332 1961 A. Hurwitz9689 2917 1963 D. B. Gillies9941 2993 1963 D. B. Gillies11213 3376 1963 D. B. Gillies19937 6002 1971 B. Tuckerman21701 6533 1978 C. Noll, L. Nickel23209 6987 1979 C. Noll44497 13385 1979 H. Nelson, D. Slowinski86243 25962 1982 D. Slowinski110503 33265 1988 W. N. Colquitt, L. Welsch132049 39751 1983 D. Slowinski216091 65050 1985 D. Slowinski756839 227832 1992 D. Slowinski, P. Gage859433 258716 1994 D. Slowinski, P. Gage1257787 378632 1996 Slowinski, P. Gage1398269 420921 1996 Armengaud, Woltman et al2976221 895932 1997 Spence, Woltman, et al3021377 909526 1998 Clarkson, Woltman, Kurowski et al6972593 2098960 1999 Hajratwala, Woltman, Kurowski et al13466917 4053946 2001 Cameron, Woltman, Kurowski et al20996011 6320430 2003 Shafer, Woltman, Kurowski et al24036583 7235733 2004 Findlay, Woltman, Kurowski et al25964951 7816230 2005 Nowak, Woltman, Kurowski et al30402457 9152052 2005 Cooper, Boone, Woltman, Kurowski et al32582657 9808358 2006 Cooper, Boone, Woltman, Kurowski et al37156667 11185272 2008 Elvenich, Woltman, Kurowski et al42643801 12837064 2009 Strindmo, Woltman, Kurowski et al43112609 12978189 2008 Smith, Woltman, Kurowski et al57885161 17425170 2013 Cooper, Woltman, Kurowski et al

M57885161 is also the largest known prime.