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"EXAMPLE. Let the given number be 84. The greatest square which measures it is 4, and the quotient is 21 which is measured by 3 or by 7, both 3 and 7 being of the form 4” – 1. I say that 84 is neither a square nor the sum of two squares either integral or fractional.

"Let the given number be 77. The greatest square which measures it is 1, and the quotient is 77 which is here the same as the given number and is measured by 11 or by 7, each of these numbers being of the form în — 1. I say that 77 is neither a square nor the sum of two squares, either in integers or fractions.

"I confess to you frankly that I have found nothing in the theory of numbers which has pleased me so much as the proof of this proposition, and I shall be glad if you will try to discover it, if only for the purpose of showing me whether I think more of my discovery than it deserves.

"Following on this I have proved the following proposition, which is of assistance in the finding of prime numbers.

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If a number is the sum of two squares prime to one another, I say that it cannot be divided by any prime number of the form 4% − 1.

"For example, add 1, if you will, to an even square, say the square 10 000 000 000, making 10 000 000 001. I say that 10 000 00 00 cannot be divided by any prime number of the form 4 – 1, and accordingly, when you would try whether it is a prime number, you need not divide by 3, 7, II etc."

(The theorem that Numbers which are the sum of two squares prime to one another have no divisors except such as are likewise the sum of two squares was proved by Euler'.)

3. Numbers (1) which are always, (2) which can never be, the sum

of three squares.

(1) The number which is double of any prime number of the form 8n – 1 is the sum of three squares (Letter to Kenelm Digby of June 1658)'.

E.g. the numbers 7, 23, 31, 47 etc. are primes of the form 87-1; the doubles are 14, 46, 62, 94 etc.; and the latter numbers are the sums of three squares.

Fermat adds "I assert that this proposition is true, though I do so in the manner of Conon, an Archimedes not having yet arisen to assert it or prove it."

Lagrange' remarks that he has not yet been able to prove the proposition completely. The form 8 - 1 reduces to one or other of the three

1 Novi Commentarii Acad. Petropol. 1752 and 1753, Vol. IV. (1758), pp. 3-40= Commentationes arithmeticae, 1. pp. 155-173.

& Oeuvres de Fermat, II. pp. 402 sqq.

"Recherches d'Arithmétique" in Berlin Mémoires 1773 and 1775=Oeuvres de Lagrange, III. p. 795.

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forms 247-1, 24n + 7, 24n + 15, of which the first two only are primes. Lagrange had previously proved that every prime number of the form 247 + 7 is of the form x2+6y. The double of this is 2x2 + 12y3, and 2x2 + 12y2 = (x + 2y)3 + (x − 2y)3 + (2y)3,

that is, 2x+12y is the sum of three squares.

The theorem was thus proved for prime numbers of the form 87- 1, wherever n is not a multiple of 3, but not for prime numbers of the form

24n-I.

Legendre', however, has the theorem that Every number which is the double of an odd number is the sum of three squares.

(2) No number of the form 24n + 7 or 4TM (24n + 7)_can_be_the_sum of three squares.

This theorem is substantially stated in Fermat's note on Dioph. v, II. We may, as a matter of fact, substitute for the forms which he gives the forms 877 and 4" (8x+7) respectively.

Legendre' proved that numbers of the form 8n + 7 are the only odd numbers which are not the sum of three squares.

4. Every number is either a square or the sum of two, three or

four squares.

This theorem is also. mentioned in the "Relation des nouvelles dé couvertes en la science des nombres" already quoted, as a case to which Fermat ultimately found himself able to apply the method of proof by descente. He says that there are some other problems which require new principles in order to enable the method of descente to be applied, and the discovery of such new principles is sometimes so difficult that they cannot be arrived at except after very great trouble.

"Such is the following question which Bachet on Diophantus admits that he could never prove, and as to which Descartes in one of his letters makes the same statement, going so far as to admit that he regards it as so difficult that he does not see any means of solving it.

"Every number is a square or the sum of two, three or four squares. "I have at last brought this under my method, and I prove that, if a given number were not of this nature, there would exist a number smaller than it which would not be so either, and again a third number smaller than the second, etc. ad infinitum; whence we infer that all numbers are of the nature indicated."

In another place (letter to Pascal of 25 September, 1654)*, after quoting the more general proposition, including the above, that every number is

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made up (1) of one, two, or three triangles, (2) of one, two, three or four squares, (3) of one, two, three, four or five pentagons, and so on ad infinitum, Fermat adds that "to arrive at this it is necessary—

(1) To prove that every prime number of the form 4n+ 1 is the sum of two squares, e.g. 5, 13; 17, 29, 37, etc.;

(2) Given a prime number of the form 4n+1, as 53, to find, by a general rule, the two squares of which it is the sum.

(3) Every prime number of the form 3n+1 is of the form x2 + 31o, .g. 7, 13, 19, 31, 37, etc.

(4) Every prime number of the form 8n + 1 or 8n + 3 is of the form x2 + 2y3, e.g. 11, 17, 19, 41, 43, etc.

(5) There is no rational right-angled triangle in whole numbers the area of which is a square.

"This will lead to the discovery of many propositions which Bachet admits to have been unknown to him and which are wanting in Diophantus.

"I am persuaded that, when you have become acquainted with my method of proof in this kind of proposition, you will think it beautiful, and it will enable you to make many new discoveries, for it is necessary, as you know, that multi pertranseant ut augeatur scientia [Bacon]."

Propositions (3) and (4) will be mentioned again, and a full account will be given in Section III. of this Supplement of Fermat's method, or methods, of proving (5).

The main theorem now in question that every integral number is the sum of four or fewer squares was attacked by Euler in the paper' (17541755) in which he finally proved the proposition (1) above about primes of the form 47 + 1; but, though he obtained important results, he did not then succeed in completing the proof. Lagrange followed up Euler's results and finally established the proposition in 17703. Euler returned to the subject in 1772; he found Lagrange's proof long and difficult, and set himself to simplify it3.

(The rest of the more general theorem of Fermat quoted above, the portion of it, that is, which relates to numbers as the sum of n or fewer n-gonal numbers, was proved by Cauchy1.)

Novi Commentarii Acad. Petropol. for 1754-5, Vol. v. (1760), pp. 3-58=Commentationes arithmeticae collectae, 1849, 1. pp. 210-233.

Nouveaux Mémoires de l'Acad. Roy. des Sciences de Berlin, année 1770, Berlin 1772, pp. 123-133 Oeuvres de Lagrange, 111. pp. 187-201: cf. Wertheim's account in his Diophantus, pp. 324-330.

"Novae demonstrationes circa resolutionem numerorum in quadrata," Acta Erudit. Lips. 1773, p. 193; Acta Petrop. I. II. 1775, p. 48; Comment. arithm. I. pp. 538-548.

Cauchy, "Démonstration du théorème général de Fermat sur les nombres polygones," Oeuvres, 11a Série, Vol. VI. pp. 320-353. See also Legendre, Zahlentheorie, tr. Maser, II. pp. 332-343.

Under this heading may be added the further proposition that

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Any number whatever of the form 8n I can only be represented as the sum of four squares, not only in integers (as others may have seen) but in fractions also, as I promise that I will prove1."

On numbers of the forms x+2y3, x2 + 3y3, x2 + 53 respectively. (1) Every prime number of the form 8n + 1 or 8n +3 is of the form x + 2y3.

This is one of the theorems enunciated in the letter of 25 Sept., 1654, to Pascal' and also in the letter of June, 1658, to Kenelm Digby'.

[In a paper of 1754 Euler says that he does not yet see his way to prove either part of the theorem. In 1759 he says he can prove the truth of the theorem for a prime number of the form 87 + 1, but not for a prime of the form 8n+3. Later, however, he proved it for prime numbers of both forms. Lagrange' also proved it for primes of the form 8n + 3.]

(2) Every prime number of the form 3n+ 1 is of the form x+3y2. The theorem is stated in the same two letters to Pascal and Digby respectively.

Lagrange naturally quotes it as "All prime numbers of the form 67+1 are of the form xa3 + 3μ3,” for of course 3n + 1 is not a prime number unless n is even.

The proposition was proved by Euler'. Lagrange proved (a) that all prime numbers of the form 12-5 are of the form x2+353, (b) that all prime numbers of the form 127-1 are of the form 3x2-y, and (c) that all prime numbers of the form 12n+1 are of both the forms +33oa and x3-3y2.

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(3) No number of the form 3n-1 can be of the form x2 + 31a3. In the "Relation des nouvelles découvertes en la science des nombres 10 Fermat says that this was one of the negative propositions which he proved by his method of descente.

1 Letter to Mersenne of Sept. or Oct. 1636, Oeuvres de Fermat, II. p. 66.

2 Oeuvres de Fermat, II. p. 313.

Ibid. II. p. 403.

4.11 Specimen de usu observationum in mathesi pura (De numeris formae 2aa+bb)” in Novi Commentarii Acad. Petrop. 1756-7, Vol. vi. (1761), pp. 185-230=Comment. arithm. I. pp. 174-192.

Novi Commentarii Acad. Petrop. 1760–1, Vol. viii. (1763), pp. 126-8=Comment. arithm. 1. p. 296.

• Commentationes arithmeticae, II. p. 607.

7 "Recherches d'Arithmétique" in Oeuvres de Lagrange, 111. pp. 776, 784. "Supplementum quorundam theorematum arithmeticorum, quae in nonnullis demonstrationibus supponuntur (De numeris formae aa+3bb)" in Novi Comment. Acad. Petrop. 1760-1, Vol. VIII. (1763), pp. 105-128 Comment. arithm. 1. pp. 287–296. Op. cit., Oeuvres de Lagrange, 111. pp. 784, 791.

10 Oeuvres de Fermat, 11. p. 431.

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(4) If two prime numbers ending in either 3 or 7 which are also of the orm 4n+ 3 are multiplied together, the product is of the form x + 53o.

This theorem also is enunciated in the letter of June, 1658, to Kenelm Digby. Fermat instances 3, 7, 23, 43, 47, 67 etc. as numbers of the kind ndicated. Take, he says, two of these, g. 7 and 23. The product 161 will be the sum of a square and 5 times another square, namely 81 + 5. 16.

He admits, however, that he has not yet proved the theorem generally: 'I assert that this theorem is true generally, and I am only waiting for a proof of it. Moreover the square of each of the said numbers is the sum of a square and 5 times another square: this, too, I should like to see proved."

Lagrange proved this theorem also. He observes that the numbers described are either of the form 20% +3 or of the form 207+7, and he proves that all prime numbers of these forms are necessarily of the form 2x2 ± 2xy + 31a. He has then only to prove that the product of two numbers of the latter form is of the form + 51a.

This is easy, for

(2xa + 2xy + 3y2) (2.x'2 + 2x'y' + 3y'2) = (2xx′ + xy′+yx' + 3yy')2 + 5 (xy' —yx')3.

6. Numbers of the forms x3-233 and 2x3-33.

Fermat's way of expressing the fact that a number is of one of these forms is to say that it is the sum of, or the difference between, the two smaller sides, i.e. the perpendicular sides, of a right-angled triangle. Like Diophantus, he "forms" a rational right-angled triangle from two numbers x, y, taking as the three sides the numbers x2 + y2, x2 − y2, 2xy respectively. The sum therefore of the perpendicular sides is x+2xy-yor (x + y)2 - 2y3, and their difference is either x-2xy-y or 2xy-(x-2), that is, either (xy)-2 or 2-(x-y).

The main theorem on the subject of numbers of these forms is, as a matter of fact, contained, not in a letter of Fermat's, but in two letters of Frénicle to Fermat dated 2nd August and 6th Sept., 1641, respectively3. It is, however, clear (cf. the letter in which Fermat had on 15th June, 1641, propounded to Frénicle a problem on such numbers) that the theorem was at any rate common property between the two.

Frénicle's two statements of the theorem are as follows:

"Every prime number of the form 87 ± 1 is the sum of the two smaller sides of a (right-angled) triangle, and every number which is the sum of the two smaller sides of a (right-angled) triangle with sides prime to one another is of the form 87 ± 1."

"Every prime number of the form 87 ± 1, or which is the product of such prime numbers exclusively, is the difference between the two smaller sides of an infinite number of primitive right-angled triangles." Op. cit., Oeuvres de Lagrange, 11. pp. 784, 788-9.

• Oeuvres de Fermat, 11. pp. 231, 235.

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