Chapter 14

Euclid’s Elements, Book VII

14.1 Definitions

Definitions (VII.1). A unit is that by virtue of which each of the things that existis called one.(VII.2). A number is a multitude composed of units.(VII.3). A number is a part of a number, the less of the greater, when it measuresthe greater;(VII.4). But parts when it does not measure it.(VII.5). The greater number is a multiple 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 of Euclid VII.4 reveals, to say that A is parts of B means thatA is a sum of numbers (or units) each of which is a part of B.

14.2 The euclidean algorithm

The first two propositions determine if two given numbers are prime 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.

502 Euclid’s Elements, Book VII

B AF H

D CG

E

Figure 14.1: 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 their greatestcommon 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 numbers a and b, assuming a > b, we construct two sequencesa2, a3, . . . , an and q1, q2, . . . , qn−1 as follows.(i) a1 = a and a2 = b.(ii) For each number i, let qi be the greatest number of times ai+1 can be subtractedfrom ai, i.e., qiai+1 ≤ ai and (qi + 1)ai+1 > ai. Set ai+2 = ai − qiai+1. Note thatai+2 < ai.Therefore the sequences terminate when an+1 = 0 for some n. This means that an

14.3 Euclid I.4-10 503

measures an−1.If an measures an−i and an−i−1, then it also measures an−i−2 since an−i−2 =an−i + qn−ian−i−1.Since an measures an and an−1, it also measures an−2, an−3, . . . , a2 = b and a1 = a.an is a common measure of a and b.On the other hand, every common measure of a and b measures a3, . . . , an.Therefore, an is the greatest common measure of a and b.

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

Notation: (A,B) = greatest common measure of A and B.

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

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

14.3 Euclid I.4-10

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

For numbers A less than B, either A measures B, or B is a multitude of thegreatest common measure of A and B.

Euclid.(VII.5). If a number be part of a number, and another be the same part of another,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 of another,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 subtracted is 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).

504 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 parts of an-other, alternately also, whatever parts of part the first is of the 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.

14.4 Numbers in proportion

Definition (VII.20). Numbers are proportional 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.

Assume a > a′, b > b′, and a : b = a′ : b′. Then a− 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 all the 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.

14.5 Commutative law of multiplication 505

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 ratio ex aequali.

If a : b = a′ : b′ and b : c = b′ : c′, then ex aequali, a : c = a′ : c′.

14.5 Commutative law of multiplication

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

3 multiplies 5 : 5 + 5 + 5,5 multiplies 3 : 3 + 3 + 3 + 3 + 3.

We read A×B as A multiplies B (or B multiplied by A), and it means the sumof A copies of B:

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 and B = pA. Consider a third number C measuring a fourthone D 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 ,

506 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;

14.6 Euclid VII.17-20 507

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.

14.6 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 numbers multiplied.

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 and A, B are the least possible, then C = nA =⇒ D = nB.

14.7 Prime and composite numbers

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

508 Euclid’s Elements, Book VII

If A and B are prime to one another, then the ratio A : 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 and A, B are least possible, then A and B are prime to oneanother.

Definitions (VII.16). And, when two numbers having multiplied one another makesome number, the number so produced be called plane, 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 called solid, and its sides are the numbers whichhave multiplied one another.(VII.18). A square number is equal multiplied by equal, or a number which is con-tained by two equal numbers.(VII.19). And a cube is equal multiplied by equal and again by equal, or a numberwhich is contained by three equal numbers.

Notation: A|B means A measures B.Euclid (VII.23-28). Let A and B be relatively prime numbers.(VII.23) If C|A, then C and B are relatively prime.(VII.24). If C and B are also relatively prime, then so are AC and B.(VII.25). A2 and B are relatively prime.(VII.26). A2 and B2 are relatively prime.(VII.27). A3 and B3 are relatively prime; so are an and bn.(VII.28). A+B and B are relatively prime, and conversely.

Note that from VII.24, VII.25 follows by putting C = A;from VII.25, VII.26 follows by putting B and A for A and B 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 of the original num-bers.

If P is prime and P |AB, then P |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.

14.8 Least common measure 509

14.8 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.

Given A, B, C, . . . , to find the least numbers A′, B′, C ′, . . . such that

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

If D = (A,B,C, . . . ), then A = 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 of A and B measuresAB. We shall write LCM(A,B).

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

A|C and B|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 by B, say, A = nB, then A has a part C corresponding to themeasuring number n.

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

14.9 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 14.1 (Gauss, DA, Art.16). A composite number can be resolved into prime factorsin only one way.

510 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 except a, b, c, etc., since no other prime can divide A which is composed of theseprimes. Similarly in this second system of factors none of the prime numbers a, 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 number appearsmore often than in the other. Let such a prime be p, which appears in one resolution mtimes and in the other n times, and let m > n. Now remove from each system the factor p,n times. As a result p will remain in one system m − n times and will be missing entirelyfrom the other. That is, we have two resolutions into factors of the number A/pn. One ofthem does not contain the factor p, the other contains it m−n times, contradicting what wehave just shown.

1Section II, Congruences of the first order, of Disquisitiones Arithmeticae begins 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 product ab cannot be divided by p. Gauss remarked that “Euclid had alreadyproved this theorem in his Elements (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 thenumbers a, b, c, d etc. can be divided by a prime p neither can their product abcd etc.

Chapter 15

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 the ratios 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 in continued 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

512 Euclid’s Elements, Book VIII

proportion, so many also fall between the numbers themselves in 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 make certain numbers, thenthe latter are also proportional.(VIII. 14). If a square measures a square, then the side also measures the side; 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, then neither 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, then thenumbers are similar solid numbers.(VIII. 22). If three numbers are in continued proportion, and the first is square, thenthe third is also square.(VIII. 23). If four numbers are in continued proportion, and the first is a cube, 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 number hasto 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.

513

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

514 Euclid’s Elements, Book VIII

Chapter 16

Euclid’s Elements, Book IX

16.1 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 form A′B′

where A′ : 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, then the 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.

16.2 Continued proportions

Euclid (IX.8). If as many numbers as we please beginning from a unit are in con-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

516 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, and a1, a2, a3, . . . , is a geometric progression, then an =an−1, and a3, a5 etc are squares, a4, a7 etc are cubes, and a7, 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 rest will also be cube.

If a is a square (cube) number, so is every an.

(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, then neither is any othersquare except the third from the unit and all those which leave out 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 greatest will not be measuredby any except those which have a place among the proportional numbers.

16.3

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 and q are distinct prime numbers, and p1p2 is not measured by q.

16.4 Infinitude of prime numbers 517

If p1, p2, . . . , pn are distinct prime numbers, then p1p2 · · · pn is the least numbermeasured by them.

If q is a prime number different from p1, . . . , 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 and b are relatively prime, there is no geometric progression a, 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 which a1 and an are relativelyprime, then a1, a2, an, a do not form a geometric progression for any number a.

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

Two given numbers a and b can be extended into a geometric progression a, b,c if and only if a measures b2.

16.4 Infinitude of prime numbers

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

Proof. Let A, B, C be the assigned prime numbers;I say that there are more prime numbers than A, B, C.

For let the least number measured by A, B, C be taken, [VII. 36]and let it be DE;let the unit DF be added to DE.

518 Euclid’s Elements, Book IX

Then EF is either prime or not.

First, let it be prime;then the prime numbers A, B, C, EF have been foundwhich are more than A, B, C.

Next, let EF not be prime;therefore it is measured by some prime number. [VII. 31]Let it be measured by the prime number G.I say that G is not the same with any of the numbers A, B, C.For, if possible, let it be so.Now A, B, C measure DE;therefore G also will measure DE.But it also measure EF .Therefore G, being a number, will measure the remainder, the unit DF :which is absurd.

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

16.5 Even and odd numbers

Definitions (IX.6). An even number is 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, is a 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

16.5 Even and odd numbers 519

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 their multitudebe 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 odd number

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 number A by multiplying the even number Bmake C;I say that C is even.For since A by multiplying B has made C,therefore C is made up of as many number equal to B as there are units in A.(Def.VII.15)And B is even;therefore C is made up of even numbers.But, if as many even numbers as we please be added together, the whole is even.(X.21)

520 Euclid’s Elements, Book IX

Therefore C is even. Q.E.D.

Euclid’s proof did not actually assume A odd. It works for any number A.

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 number is that which is measured by aneven number according to an even number.(IX.9). An even-times odd number is 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 is even-times even only.

These are powers of 2.

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.

16.6 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 = a1a2 − a1 (an+1 − a1)

16.7 Perfect numbers 521

Proof.

a2a1

=a3a2

= · · · = anan−1

=an+1an

=⇒ a2 − a1a1

=a3 − a2

a2= · · · = an − an−1

an−1=

an+1 − anan

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

a1 + a2 + · · ·+ an−1 + an=

an+1 − a1a1 + a2 + · · ·+ an−1 + an .

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

16.7 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, suppose Mn = 1+2+22 + · · ·+2n−1 is a prime number,then 2n−1 ·Mn is a perfect number. Note that

1 + 2 + 22 + · · ·+ 2n−1 = 2n − 1.

n 2n − 1 perfect number 2n−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.

522 Euclid’s Elements, Book IX

(1) The quest of prime numbers of the form Mn := 2n − 1. It is easy to see thatn must be a prime if Mn is a prime. The converse needs not be true. To date (May,2014), there are only 48 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 an odd 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 least 4 distinct prime divisors.(ii) Current record: P. Nielsen, Odd perfect numbers have at least 9 distinct

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

16.7.1 Euler’s proof

Euler made use of the sum-of-divisors function σ(P ), which has the multiplicativeproperty: σ(PQ) = σ(P )σ(Q) for relatively prime integers P and Q. In terms ofthe function σ, a number n is perfect if and only if σ(n) = 2n.

Let P be an even perfect number. Write P = 2n−1 · Q for n − 1 ≥ 1 and Qconsisting only of odd divisors. Since 2n−1 and Q 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 that 2n − 1 is a divisor ofQ, so is Q

2n−1 . Since σ(Q) is the sum of all divisors of Q. This equation showsthat Q

2n−1 must be 1, and Q has no more divisors apart from Q and 1. From this weconclude that Q = 2n − 1 and is a prime number (since its only divisors are Q and1). The even perfect number is P = 2n−1(2n − 1) constructed in Euclid IX.36.

1The latest one M57885161 has 17425170 digits, and was discovered in 2013. The correspondingperfect number has 34850339 digits.

16.7 Perfect numbers 523

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.