1. The Greeks

When students first come across unique factorization in integers, their mainproblem is not understanding the concept but rather accepting that thisproperty, which seems obvious from the experience gathered by years of com-puting, requires a proof at all. It is therefore quite remarkable how closeEuclid (325 – 265 BC)1 came to stating and proving unique factorization.

1.1 Pythagoras

In this section we will discuss three contributions of the Pythagoreans toGreek2 mathematics that have a connection with unique factorization: theirrationality of

√2, perfect numbers, and Pythagorean triples. In the next

section we will see that these topics were taken up again (or at least lefttraces) in Euclid’s Elements.

Incommensurability

It is believed that the irrationality of√

2 was already known to the Pythago-reans, a society founded by Pythagoras (569 – 475 BC) in Italy; legends thatPythagoras himself or Hippasus (500 – ?? BC) discovered irrationality whilestudying the square or the pentagram (which would prove the irrationalityof√

5 ) seem to be just that: legends.Aristotle (384 – 322 BC) [5] gave a proof for the irrationality of

√2, which

is usually credited to the Pythagorean school, and which, when translatedinto our mathematical language, goes like this: Assume that

√2 = m/n,

where the fraction is written in lowest terms; then 2n2 = m2. Since 2 dividesthe left hand side, it must divide m2. But then m is even, say m = 2a. Thisgives 2n2 = 4a2 and n2 = 2a2. Repeating the argument now shows that n iseven as well, contradicting the assumption that m/n was written in lowestterms.1 The dates given here should be regarded as crude approximations; our knowledge

about the biographies of most Greek mathematicians is rather scant.2 I’m using the term “Greeks” in the broadest sense; it includes mathematicians

who lived in the Mediterranean area such as in Greece as well as in Italy (such asPythagoras), Lybia (Theodorus), or Alexandria in Egypt (Euclid, Diophantus),to name a few.

2 1. The Greeks

Perfect and Amicable Numbers

Another pastime of the Pythagoreans was studying perfect and amicablenumbers. A number is called perfect if it is the sum of its proper divisors.Thus 6 = 1+2+3 and 28 = 1+2+4+7+14 are perfect. The Pythagoreansknew only one recipe for constructing perfect numbers: if 2p − 1 is prime,then N = 2p−1(2p−1) is perfect. The proof consists in showing that the onlyfactors of N are 2r and 2r(2p − 1) for r = 0, 1, . . . , p − 1. For us, this is atrivial consquence of unique factorization.

Pythagorean Triples

Finally let us discuss Pythagorean triples. These are solutions of the equa-tion x2 + y2 = z2 in natural numbers. The Pythagoreans had rules thatallowed them to construct as many triples as they wanted; Proclus creditsthe Pythagoreans with the formula

x = 2n+ 1, y = 2n2 + 2n, z = 2n2 + 2n+ 1,

and Plato withx = 2n, y = n2 − 1, z = n2 + 1.

Finding infinitely many Pythagorean triples has nothing to do with uniquefactorization; this only becomes relevant for anwering the question how tofind all solutions of a diophantine equation such as x2 + y2 = z2.

1.2 Theodorus

There is a famous dialogue written by Plato (427 – 347 BC) (see Wasserstein[169], or Knorr [86, pp. 62–64] for the Greek original) in which Theodorus ofCyrene (465 – 398 BC) is said to have proved the irrationality of the squareroots of nonsquares from 3 up to 17; here is the standard translation:

[Theodorus] was proving to us a certain thing about square roots,I mean the side of a square of three square units and of five squareunits, that these roots are not commensurable in length with the unitlength, and he went on in this way, taking all the separate cases up tothe root of seventeen square units, at which point, for some reason,he stopped.

Before we discuss the implications, let us first explain the notion of commen-surability. Two line segments A and B are said to be commensurable if theboth can be measured with a common measure, that is, if there is a line seg-ment C such that both A and B are (integral) multiples of C. When expressedin modern terms, this means that two line segments are commensurable ifthe ratio of their lenghts is a rational number.

1.2 Theodorus 3

The simplest example of incommensurable lines are the side and the di-agonal of a square: their lengths have ratio

√2. Slightly more involved and

perhaps closer to the Pythagoreans’ hearts and minds is the example of theside and any diagonal of the regular pentagon, which corresponds to theirrationality of

√5−12 .

Plato’s dialogue contains two important pieces of information: first, theirrationality of such numbers was known to Theodorus. Second, the fact thatPlato does not credit the irrationality of

√2 to him shows that this must

have been known before.In particular, it is not known how Theodorus stated (let alone proved)

his claims; it seems difficult to state the general result in Euclid’s geometriclanguage (but see the discussion of Euclid’s Proposition VIII.7 below).

The Pythagorean’s proof that√

2 is irrational (in the sense of thePythagoreans) was included as Proposition 117 in Euclid’s Book X, but isbelieved to be a later addition; current editions of Euclid’s Elements give“only” 115 propositions in Book X.

One reason why this proof is believed to go back to the Pythagoreans isthat it only uses the “theory of the even and the odd”, as the rudiments ofnumber theory were called at the time. Proofs for the intermediate steps inthe above proof can be found in Euclid’s Book IX:

• Proposition 28. If an odd number is multiplied by an even number, thenthe product is even.

• Proposition 29. If an odd number is multiplied by an odd number, thenthe product is odd.

• Proposition 30. If an odd number measures an even number, then it alsomeasures half of it.

In fact, Proposition IX.29 shows that the product of two odd numbers is odd;thus q2 = 2p2 implies that q must be even.

It is easy to see that the same argument can be used to prove the ir-rationality of

√p for primes p if we can show that pn2 = m2 implies that

p | m. This is basically the content of Proposition VII.30 of Euclid. The sameargument actually works for general squarefree numbers, and with a slightmodification for all nonsquares. In fact, for proving the irrationality of

√12

we cannot deduce from 12n2 = m2 that 12 | m; we could, however, simplyremark that

√12 = 2

√3 is irrational because

√3 is.

This was, however, almost certainly not what Theodorus did in his proofsfor the irrationality of sqare roots up to

√17. In fact, Knorr [86] has con-

vincing arguments that suggest that the standard translation of the dialogueis incorrect, and that “at which point, for some reason, he stopped” shouldbe translated as “at which point, for some reason, he ran into problems”. Ifthis translation is correct, then the proofs by Theodorus cannot be based onproperties of primes, like some modern proofs are. In fact, Knorr has workedout proofs that would explain why Theodorus ran into problems when he

4 1. The Greeks

arrived at 17; even if this reconstruction should not be correct, it is at leastbeautiful mathematics based on mathematics known to the Pythagoreans.

In fact, Knorr suggests that Theodorus proved the irrationality of√

3 etc.by using known properties of Pythagorean triples, the most important being

Proposition 1.1. If A2 +B2 = C2 and C is even, then so are A and B.

Using the modern notion of congruences, this is a rather trivial exercise.Knorr explains that this should have been within reach for the Pythagoreans.

Now assume that√

3 is rational, say 3 = A2 : B2; assume that A : B iswritten in lowest terms. Then 3B2 = A2, or A2 +B2 = 4B2. By Proposition1.1, this implies that A and B are even, which contradicts our assumptions.

TTTT

√3

1

2TTTT

A

B

2B

Knorr also shows how to generalize this gem of a proof to more generalnumbers: assume that p = 4n + 3 is prime, and that p = A2 : B2, whereA : B is written in lowest terms. Then the fact that (p−1

2 )2 + p = (p+12 )2

shows that A2 + [(2n+ 1)B]2 = [(2n+ 2)B]2. Proposition 1.1 implies that Aand (2n + 1)B are even, hence A and B are even, and this contradicts ourassumptions.

What happens for primes p = 4n + 1? If√p = A2 : B2 with A : B in

lowest terms, then there is a right triangle with sides 2nB, A and hypotenuse(2n+ 1)B. If B is even, then so is A, and we get a contradiction. Thus B isodd, and this implies that A must be odd, too. Thus we get a contradictionfor odd n = 2k+1 (i.e., for primes p = 8k+5) from the following observation:

Proposition 1.2. If there is a right triangle with sides A, B and hypotenuseC, and if B and C are odd, then B is divisible by 4.

This is again a trivial consequence of looking at A2 +B2 = C2 modulo 8,but certainly within reach of the Pythagoreans.

If n is even, i.e., if p is a prime of the form 8n+1 (like 17), then the aboveapproach to proving the irrationality of

√p will fail. Knorr observes that it

is possible to prove this using “congruences modulo 3” instead of the theoryof the odd and the even.

1.3 Euclid

Euclid’s Elements were an attempt to axiomatize the mathematical knowl-edge of the Greeks. The geometric language he used for describing what wetoday would regard as number theory makes reading and understanding the

1.3 Euclid 5

Elements a difficult task for today’s readers. In the following we will quotesome parts of the Elements that are related the problem of unique factoriza-tion, and then translate them into modern language and discuss their content.

Euclid’s Language

In order to better understand the limitations of Euclid’s language, let us nowdiscuss one of the best known proofs in the Elements, namely his proof thatthere are infinitely many primes (Proposition IX.20):

Prime numbers are more than any assigned multitude of prime num-bers.

Observe first that Euclid does not claim that there are infinitely manyprimes, but only that every finite set of primes can be enlarged. This choiceof language is not an accident: it reflects the Greeks’ extreme caution whenit comes to the notion of infinity. Since paradoxa connected with infinitywere not properly understood (and would not be properly understood forthe next two millenia), the Greeks refused to consider actual infinities andpreferred to talk about potential infinities only: whereas we think of a line inthe Euclidean plane as an infinite object, for Euclid lines were line segmentsthat could be extended as far as desired.

Next let us discuss Euclid’s definition of primes. First, a number is a(proper) multiple of the unit. Thus Euclid distinguishes between the unit1 and the numbers 2, 3, 4, . . . . “To measure” and “be measured by” canbe translated as “to properly divide” and “to be divisible properly by”. Inparticular, the number 6 is measured by 1, 2 and 3, hence is perfect because1 + 2 + 3 = 6. Euclid’s Definition VII.11 says that “A prime number isthat which is measured by a unit alone”; thus 1 is not a prime because it isnot a number. Definition 13 says that “A composite number is that whichis measured by some number”. Thus 1 is not composite because it is onlydivisible by the unit, which is not a number.

Now let us see how Euclid proves his Proposition IX.20 according to whichevery finite list of primes can be enlarged:

Let A, B, and C be the assigned prime numbers.I say that there are more prime numbers than A, B, and C. Take theleast number DE measured by A, B, and C. Add the unit DF to DE.

A B C•E

•D•F

Then EF is either prime or not.First, let it be prime. Then the prime numbers A, B, C, and EF havebeen found which are more than A, B, and C.

6 1. The Greeks

Next, let EF not be prime. Therefore it is measured by some primenumber. Let it be measured by the prime number G. I say that G isnot the same with any of the numbers A, B, and C.If possible, let it be so.Now A, B, and C measure DE, therefore G also measures DE. Butit also measures EF. Therefore G, being a number, measures theremainder, the unit DF, which is absurd.Therefore G is not the same with any one of the numbers A, B,and C. And by hypothesis it is prime. Therefore the prime numbersA, B, C, and G have been found which are more than the assignedmultitude of A, B, and C.Therefore, prime numbers are more than any assigned multitude ofprime numbers.

The first thing that strikes us as weird is that Euclid does not form thenumber ABC + 1 as we would, but rather takes the least number divisibleby A, B and C (in modern terms: the lowest common multiple of A, B, C)and adds the unit. The reason is that Euclid distinguishes between numbers,plane numbers (products of two numbers) and solid numbers (products ofthree numbers); moreover, he never adds e.g. numbers and plane numbers,so you will look in vain for an expression of the form A ·B + 1 in Euclid.

Apart from the unfamiliar language and the fact that Euclid gives theproof “symbolically” for a list of three primes only, the proof is the same asthe one known from textbooks.

Book VII

Book VII contains several fundamental properties of numbers:

• Proposition 1. When two unequal numbers are set out, and the less iscontinually subtracted in turn from the greater, if the number which isleft never measures the one before it until a unit is left, then the originalnumbers are relatively prime.

• Proposition 2. To find the greatest common measure of two given numbersnot relatively prime.

• Proposition 24. If two numbers are relatively prime to any number, thentheir product is also relatively prime to the same.

• Proposition 29. Any prime number is relatively prime to any numberwhich it does not measure.

• Proposition 30. If two numbers by multiplying one another make somenumber, and any prime number measure the product, it will also measureone of the original numbers.

• Proposition 31. Any composite number is measured by some prime num-ber.

• Proposition 32. Any number is either prime or is measured by some primenumber.

1.3 Euclid 7

Proposition VII.2 proclaims the existence of the gcd of two numbers thatare not coprime; as was customary in the Elements, such existence proofsvia construction are stated as problems. The construction of the gcd wasperformed by the “Euclidean Algorithm” (which can already be found in thework of Aristotle; actually, it is believed that Euclid “only” compiled andstreamlined the work of other mathematicians such as Eudoxus (408 – 355BC) or Theaetetus (417 – 369 BC), to name but two.

Proposition VII.1 seems to be a special case of VII.2, because it claimsthat gcd(a, b) = 1 if the Euclidean algorithm applied to a and b gives 1 asthe last remainder. For Euclid, however, 1 was not a number, which meansthat he has to distinguish two cases: gcd(a, b) is the unit (VII.1), or it is anumber (VII.2).

Proposition VII.24 claims that if a, c and b, c are coprime, then so are aband c. This proposition and its consequences (for example, Euclid uses it toderive that gcd(a, b) = 1 implies gcd(a2, b) = gcd(a2, b2) = 1, and similarlyfor cubes) are all over the place in Book VIII; it may be seen as Euclid’s“fundamental theorem” of number theory.

Proposition VII.29 states that if a prime p does not divide an integera then gcd(p, a) = 1, and Proposition VII.30 is the corner stone on whichwe build the fundamental theorem today: a prime dividing a product mustdivide one of its factors.

Next, Proposition VII.31 states that any composite number is divisibleby a prime. Euclid’s proof proceeds as follows: Since the given number A iscomposite, we have A = aB. If B is prime, then we are done; if not, thenB = bC and thus A = abC. Repeating this process will yield a prime dividingA. For if not,

then an infinite sequence of numbers measures the number A, eachof which is less than the other, which is impossible in numbers.

Again we can observe that Euclid only uses products of up to three factorsin his proof. There is, however, a difference to the proof of IX.20 discussedabove: there, Euclid was talking about the smallest number measured by A,B, C, and it is clear that he might just as well have used the smallest numbermeasured by A, B, C, D. Here, on the other hand, Euclid actually looks atproducts, and although he knows and admits that products of more thanthree factors exist, his language would not allow him to talk about them.

Euclid’s Proposition VII.32 states that any number is either prime ordivisible by a prime. This is a modest step towards a formulation of theexistence of a prime factorization: if the number N is divisible by the primep, then N/p is either prime or divisible by a prime; continuing in this manner,we get a prime factorization of N after finitely many steps. This construction,however, was not given by Euclid.

8 1. The Greeks

Book VIII

Book VIII deals with plane and solid numbers, as well as with geometricsequences. By far the most interesting propositions for us are the following:

• Proposition 7. If there are as many numbers as we please in continuedproportion, and the first measures the last, then it also measures thesecond.

• Proposition 9. If two numbers are relatively prime, and numbers fallbetween them in continued proportion, then, however many numbers fallbetween them in continued proportion, so many also fall between each ofthem and a unit in continued proportion.

These are statements that contain much more than meets the eye at first.Let us first discuss Proposition VIII.7. To say that “numbers fall betweenthem in continued proportion” means that they are the beginning and theend of a geometric sequence: a = a0, a1, . . . , an = b, and the ratios aj+1 : aj

are all equal. Proposition VIII.7 then claims that if a, b, . . . , h is a geometricsequence, and if a | h, then a | b.

We can use this to prove the irrationality of√m for nonsquares n as

follows: assume that m = p2/q2, where p/q is in its lowest terms. Thenq2, pq, p2 = mq2 form a geometric sequence, and so does q, p,mq. Since thefirst term divides the last, we have q | p, hence m is a perfect square. Thesame reasoning works with nth powers instead of squares. Whether such aproof was acceptable to Euclid is doubtful; but there is nothing wrong withthis proof from the point of view of the contemporaries of Fermat and Euler,who did not have problems with identifying ratios p : q and fractions p/q, orwith adding solid numbers to ordinary numbers.

Now consider Proposition VIII.9, and let a and b be two coprime numbers.Let us first discuss the special case where n = 2: here the geometric sequenceis a, c, b, and we have (replacing Euclid’s ratios by fractions) c

a = bc , or,

equivalently, ab = c2. Euclid then claims that there are geometric sequencesof length 2 starting with 1 and ending with a and b, respectively. But if 1, r, ais a geometric sequence of integers, then a = r2 must be a perfect square.

Thus the special case n = 2 of Euclid’s Proposition VIII.9 may be trans-lated into modern terminology as follows: if ab = c2 and gcd(a, b) = 1, thena and b are perfect squares.

The complete proof of Euclid’s Proposition VIII.9 is quite involved. Back-tracking the propositions he uses we find that it is based on the Euclideanalgorithm via VIII.9 → VIII.2 → VII.20 → VII.4 → VII.2; the same holdsfor VIII.7.

Book IX

Book IX deals with squares, cubes and geometric sequences, the theory ofthe odd and even, and with number theory proper.

1.3 Euclid 9

• Proposition 1. If two similar plane numbers multiplied by one anothermake some number, then the product is square.

• Proposition 2. If two numbers multiplied by one another make a squarenumber, then they are similar plane numbers.

• Proposition 13. If as many numbers as we please beginning from a unit arein continued proportion, and the number after the unit is prime, then thegreatest is not measured by any except those which have a place amongthe proportional numbers.

• Proposition 14. If a number be the least that is measured by prime num-bers, it will not be measured by any other prime number except thoseoriginally measuring it.

• Proposition 20. Prime numbers are more than any assigned multitude ofprime numbers.

• Proposition 36. If as many numbers as we please beginning from a unitare set out continuously in double proportion until the sum of all becomesprime, and if the sum multiplied into the last makes some number, thenthe product is perfect.

Definition VII.21 states that similar plane numbers are those which havetheir sides proportional. This means that m and n are similar plane numbersif they can be written in the form m = ab and n = cd with a : b = c : d. Thenumbers 8 and 18, for example, are similar plane numbers because 8 = 2 · 4and 18 = 3 · 6; observe that their product 8 · 18 = 362 is a square.

Proposition IX.2 can also be used to give a proof for the following specialcase of Proposition VIII.9:

Proposition 1.3. Assume that a and b are coprime numbers such that ab isa perfect square. Then a and b are perfect squares.

Proof. By Euclid’s Proposition IX.2, a and b are similar plane numbers, i.e.,a = rs and b = tu with r : t = s : u. The last equation implies ru = st(VII.19). Since a and b are coprime, so are r and t (VII.23). From (VII.21)and (VII.20) we may now conclude that r = s or r | s. By symmetry, we alsohave s = r or s | r, hence r = s, and, similarly, t = u.

Euclid’s Proposition IX.14 says that the lowest common multiple of primeshas no prime factors different from these primes. This is a special case of thetheorem of unique factorization. The problem here is that although Euclidshows that primes not occurring in the original factorization do not divide theproduct, it is not claimed that the primes that occur do so with well definedmultiplicity, that is: it does not exclude the possibility that e.g. pq2 = p2q.For us, this is not really essential since this problem can easily be solved bycanceling, but Hendy [67] observes that “Euclid’s methods cannot be adaptedto prove unique factorization for numbers containing square factors”.

Proposition IX.13 deals with numbers beginning from unit in continuedproportion, i.e., with a geometric sequence 1, p, p2, . . . , pn, where the number

10 1. The Greeks

p after the unit is a prime; then the greatest number pn is divisible only bythe numbers preceding it in this sequence, namely by 1, p, . . . , pn−1.

Proposition IX.36 states that numbers of the form 2n−1(2n−1) are perfectif 2n − 1 is prime: the numbers in double proportion beginning from a unitare 1, 2, 22, . . . , 2n−1 (of course here I mean 1, 2, 4, . . . , not 1, 2, 2 · 2, . . . asthe modern notation suggests. Euclid would only add numbers 1+2+4+ . . .,but not 1 + 2 + 2 · 2 + . . .). If their sum 2n − 1 is a prime, then the productof the prime with the last number 2n−1 is perfect. For the proof, Euclid hasto show that the only proper divisors of 2n−1(2n − 1), where p = 2n − 1 isprime, are 1, 2, 4, . . . , 2n−1, p, 2p, 4p, . . . , 2n−1p.

Listing all divisors of a number was also important for finding amicablenumbers; these are pairs m, n of numbers such that the sum of the properdivisors of m equals n and vice versa. The smallest pair of amicable numbersconsists of 220 and 284: in fact, the divisors of 220 are 1, 2, 4, 5, 10, 11, 20,22, 44, 55, 110, which add up to 284; similarly, the divisors 1, 2, 4, 71 and142 add up to 220.

Book X

Euclid’s Book X deals with the classification of irrational “numbers” (calledmagnitudes by Euclid). We may safely think of a magnitude as the lenghthor the area of some geometric figure (but see the Notes below).

Euclid calls two magnitudes commensurable if their quotient is rational.Thus the side and the diagonal of a square are not commensurable. On theother hand, the squares built on the side and the diagonal of a square, re-spectively, are commensurable: Euclid calls such magnitudes commensurablein square.

In Definition 3, Euclid says that a magnitude is “rational” if it is com-mensurable or commensurable in square with a chosen unit. In particular,the diagonal of a unit square is “rational” in this sense. A better translationof the original Greek term would be “enunciable” (aussprechbar in German,enoncable in French). Book X studies the “irrationality” of magnitudes thatoccur in Platonic solids, which are then discussed in Book XIII.

In Book X, Euclid also constructs Pythagorean triples:

• Lemma 1, Proposition 29. To find two square numbers such that theirsum is also square.

Proof. Set out two numbers AB and BC, and let them be either both evenor both odd. Then since, whether an even number is subtracted from aneven number, or an odd number from an odd number, the remainder is even,therefore the remainder AC is even.

Bisect AC at D. Let AB and BC also be either similar plane numbers, orsquare numbers, which are themselves also similar plane numbers.

Now the product of AB and BC together with the square on CD equalsthe square on BD. And the product of AB and BC is square, inasmuch as

1.4 Diophantus 11

it was proved that, if two similar plane numbers multiplied by one anothermake some number, the product is square. Therefore two square numbers,the product of AB and BC, and the square on CD, have been found which,when added together, make the square on BD.

And it is manifest that two square numbers, the square on BD and thesquare on CD, have again been found such that their difference, the productof AB and BC, is a square, whenever AB and BC are similar plane numbers.But when they are not similar plane numbers, two square numbers, the squareon BD and the square on DC, have been found such that their difference, theproduct of AB and BC, is not square.

In order to see what Euclid is doing, put u = AB, v = BC; in the firstparagraph he chooses u and v with the same parity.

Since D bisects AC, we have CD = u−v2 and BD = CD + v = u+v

2 . NowEuclid observes the identity uv + (u−v

2 )2 = (u+v2 )2. Thus in order to find a

Pythagorean triple, he has to make the product uv a square. By PropositionIX.2, this is the case if and only if u and v are similar plane numbers.

Thus if we put u = mp2 and v = mq2 with p ≡ q mod 2, then Euclid’s con-struction provides us with the Pythagorean triple mpq, 1

2m(p2−q2), 12m(p2 +

q2).What then is Euclid doing in the last paragraph? He says that two square

numbers, say r2 and s2, have been found such that their difference (r−s)(r+s)is a square if and only if the factors r+s and r−s are similar plane numbers.Following up on this remark leads to a complete classification of Pythagoreantriples: from r+s = mp2 and r−s = mq2 we get r = mp2−q2

2 and s = mp2+q2

2 .

1.4 Diophantus

It is believed that Diophantus lived around 250 AD; apart from this we thinkthat he lived for 84 years, since a puzzle given by Metrodorus around 500AD says

his boyhood lasted 1/6th of his life;he married after 1/7th more;his beard grew after 1/12th more,and his son was born 5 years later;the son lived to half his father’s age,and the father died 4 years after the son.

This gives rise to a linear equation in Diophantus’ age x (much simpler thananything Diophantus has done) with x = 84 as the solution.

Diophantus’ main claim to fame rests on his book “Arithmetika”, whichconsists of 13 parts. Six of them were known since Fermat’s times, anotherfour have been discovered in Arabic translation. In these books, Diophantussolves “indeterminate equations”: a determinate equation is an equation that

12 1. The Greeks

determines the solutions, like x2−2x+3 = 0; an example of an indeterminateequation is x2 + y2 = 1, which has many rational solutions.

Diophantus invented algebraic notation: he had a symbol for one unknownx, and other symbols for x2, . . . , x6. This bold step was unsurpassed untilViete (1540–1603) improved upon it by using vowels for unknown and conso-nants for known quantitites; Descartes later introduced the modern variantwhere unknowns are denoted by x, y, z and known quantities by a, b, c . . . .

Diophantus had a notion of number that differed from Euclid’s: the num-bers Diophantus works with are positive rational numbers, not just propermultiples of the unit. Moreover, while Euclid would never add a number anda plane number, expressions such as 2x− 4 are all over the place in the workof Diophantus.

In the problems of Diophantus, unique factorization played no role at all:Diophantus was completely satisfied once he had found a single solution toa given problem, and although he mentions the existence of infinitely manysolutions to a few problems, the question of finding all solutions never enteredhis mind; such questions were apparently asked for the first time by the Arabsin the 10th century. In particular, unique factorization is regularly involvedwhen it comes to proving that certain diophantine equations do not have anynontrivial solutions; note, however, that for such problems the knowledge ofEuclid’s Proposition VII.30 is often sufficient.

To divide a given square into a sum of two squares

This is one of the most famous problems by Diophantus (book II, problem 8).It became important when Fermat, in his copy of Diophantus’ Arithmeticaedited by Bachet, noted that he had this wonderful proof that cubes can’tbe written as a sum of two cubes, fourth powers not as a sum of two fourthpowers, and so on, but that the margin of this book was too small to containit. (This observation was later published by his son Samuel; Fermat publiclyonly claimed to have proofs for exponents 3 and 4).

Diophantus’ solution proceeds as follows: since his notation allows onlyone unknown, he cannot treat the general case, so he starts by assumingthat the given square is 16; letting x2 denote one of the squares whose sumis 16, the other square must be 16 − x2. He writes this square in the form(2x − 4)2, and then has to solve 16 − x2 = (2x − 4)2 = 4x2 − 16x + 16,which gives 5x2 = 16x, hence 5x = 16 (Diophantus has no problems withcanceling x since he doesn’t know 0). Now x = 16

5 , so 2x − 4 = 125 , and in

fact 25625 + 144

25 = 40025 = 16.

Diophantus is aware of the fact that his method produces many moresolutions: he writes

I form the square from any number of x minus as many units as thereare in the side of 16.

1.4 Diophantus 13

The side of 16 is the square root of 16 (think of 16 as the area of square),so in modern terms his statement means: instead of 2x− 4, the substitutionmx−4 will produce a rational solution (of course m has to be chosen positiveand rational).

Solution with modern algebraic notation. Now let us solve the problemx2 + y2 = a2, where a is a given positive rational number, using modernnotation. Diophantus’ idea is to write y = mx−a (for us, mx+a would workequally well), which gives a2 = x2 +m2x2−2amx+a2, that is, x2(1+m2) =2amx. Thus either x = 0 or x = 2am

1+m2 . Note that the substitution y = mx+awould have given x = − 2am

1+m2 , which, for a and m positive rational numbers,would produce negative values. Thus Diophantus had good reasons for pickingy = mx− a instead: he knows how to subtract a number from a larger one,but he does not know negative numbers.

Summary

The number theory of the Pythagoreans consisted basically of studying thetheory of odd and even numbers. Euclid’s attempts to put the Pythagoreanresults on an axiomatic basis led him to the realization that everything heneeded could be derived from the existence of the greatest common divisor,that is, the Euclidean algorithm.

What we know as the fundamental theorem of arithmetic played no rolewhatsoever in the Elements. Although IX.14 seems to come close, this was aside result related to studying perfect numbers, and anything but fundamen-tal for Euclid: in fact, it is used nowhere at all.

It also should be noted that the Greeks had not yet realized that theconcept of uniqueness can be used to prove results. It is not even clear towhich extent Euclid was aware of the fact that certain constructions he givesare unique. Take, for example, his definition of the center of a circle in BookI:

• Definition 15. A circle is a plane figure contained by one line such thatall the straight lines falling upon it from one point among those lyingwithin the figure equal one another.

• Definition 16. And the point is called the center of the circle.

Modern mathematicians realize immediately that the first thing we shoulddo is to show that the center of a circle is unique. This can be done usingEuclid’s techniques, and even follows from his construction of the center ofa given circle in Proposition III.1, but it is not mentioned anywhere in theElements.

On the other hand, Propositions X.42 – X.47 are all results concerningthe uniqueness of certain points.

14 1. The Greeks

Notes

Heller [66] suggested that Theaetetus employed a geometric proof, while vander Waerden [167] presented a different technique based on explicit solutionsof x2 − py2 = a for small values of a. For other discussions on this topic, seeGiacardi [55] and McCabe [123].

Apparently it was Heath [64, vol. II, p. 402] who claimed that Euclid’sProposition IX.14 is basically the unique factorization theorem (see also Art-mann [11] and Kline [84, p. 79–80]): in his comments following this proposi-tion, he writes

In other words, a number can be resolved into prime factors in onlyone way.

Rashed [139, p. 290] remarks that “we can simply note the absence ofany formulation or proof of the existence of the factorization of an integerinto prime factors”. More than 2000 years after Euclid, Gauss also skips theproof since the existence of prime factorizations “is clear from elementaryconsiderations”.

Knorr [87] holds the view that Euclid’s “treatment of just three factors isa stylistic tactic not intended to limit the generality of the results”.

Laubenbacher & Pengelley [109, p. 182–183] claim that the last paragraphin Euclid’s proof of Lemma 1 in Prop. X.29 amounts to a classification of allPythagorean triples.

Magidin & McKinnon [119] claim that in Euclid’s “discussion of Pytha-gorean triples a key step uses the fact that if the product of two relativelyprime integers is a square then each of the integers must itself be a square, aresult which requires unique factorization”.

The Euclidean idea of a magnitude should be separated from our notionof length, area or volume. The Greeks did not think of the length of a linesegment as some kind of number but as something that may be comparedwith other lengths. Thus modern readers will look in vain for a formula givingthe area of a triangle; rather Euclid constructs a square with the same areaas the given triangle, or, more generally, that of any given polygon (Prop.II.14). The most famous works of Archimedes (287 – 212 BC) follow the samepattern: the area of a segment of a parabola is given in terms of a specificinscribed triangle, and the volume of the sphere in terms of the volume ofthe circumscribed cylinder.

After the decline of Greek mathematics, Euclid’s Elements became tooadvanced and could not be used as a textbook anymore. A very successfulreplacement was a book by Boethius (480 – 524) (see Friedlein [53]). Euclid’salgorithm is described, but its correctness is not proved anymore. In one ofhis examples, Boethius shows that the common measure of XXI and VIIII isIII.

1.4 Diophantus 15

Exercises

1.1 Rewrite the Pythagorean proof of the irrationality of√

2 in Euclid’s language.Note that an equation like 2n2 = m2 has to be replaced either by 2n2 = 1m2

or n2 + n2 = m2 since Euclid does not add plane and solid numbers.

1.2 Prove that√

17 is irrational using the technique of Theodorus-Knorr, andusing congruences modulo 3. Generalize this proof to all primes p = 3n + 2.

1.3 The following irrationality proofs are “modern”. This here is taken from Fine[50], and is similar to a proof given by Dedekind [28]:

Assume that n is not a perfect square, and that√

n = p/q with p minimal.1. There is an integer k with k − 1 < p

q< k.

2. Show that pq

= rs

for the natural numbers r = (k− pq)p and s = (k− p

q)q.

3. Show that r < p.

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