228 HISTORY OF THE THEORY OF NUMBERS. [CHAP. VI 



take half of the remainder as the exponent of any prime 4n + 1. [Since 



2w + 1 = (2a + 1)(26 + l)(2c + I)"-, 



by D.] For w = 7, 15 = (2 + l)(2-2 + 1), and pq 2 answers the question. 



(F) To find a number which shall be the sum of two squares in any 

 assigned number w of ways. For w = 10, set 2w = 2-2-5. Subtracting 

 1 from each prune factor, we get 1, 1, 4. Take three primes of the form 

 4n + 1; for example, 3, 13, 17. The number sought is the product of two 

 of these by the fourth power of the third. 



(G) Conversely, to find in how many ways a given number, say 325, 

 is the sum of two squares, consider its prime factors of the form 4n + ! 

 Since 325 = 5 2 -13, we take f {2-1 + 2 + 1 + 1} = 3. Then 325 is the 

 sum of two squares in three ways. For three exponents a, b, c, the number 

 of ways is k/2 if k = (a + 1)(6 + l)(c + 1) is even, but is (k - l)/2 if k 

 is odd. 



(H) To find an integer which is the hypotenuse of any assigned number 

 w of right triangles, and which if increased by a given number a becomes 

 a square. The question is difficult. If w = a 2, 2023 and 3362 satisfy 

 the conditions, as do also an infinitude of numbers. 



That no number 4n 1 is a square or a sum of two rational squares 

 was communicated to Descartes March 22, 1638, as having been proved 

 by Fermat. Descartes 11 proved this for integral squares by observing 

 that a square is of the form 4k or Sk + 1. 



Fermat 12 stated that he had proved that a number is neither a square nor 

 the sum of two squares, integral or fractional, if its quotient by the largest 

 square dividing it contains a prime factor 4n 1 ; and that x 2 + y 2 is 

 divisible by no prime 4n 1 if x and y are relatively prime. 



Fermat (Oeuvres, II, 213) stated the contents of A, B, D, E in a letter 

 to Mersenne, Dec. 25, 1640. Frenicle, in a letter to Fermat (ibid., 241), 

 Sept. 6, 1641, proposed the problem to find the least number in F. T. 

 Pepin 13 noted that this problem and D are answered by the theory of 

 quadratic forms. 



Fermat 14 called the theorem that every prime 4n + 1 is a sum of two 

 squares [cited henceforth as Girard's 9 theorem] the fundamental theorem 

 on right triangles. He 16 stated that he possessed an irrefutable proof. 

 Elsewhere he 16 stated that his proof was by the method of indefinite descent : 

 ' If a prime 4n + 1 is not a sum of two squares, there exists a smaller 

 prune of the same nature, then a third still smaller, etc., until the number 

 5 is reached," thus leading to a contradiction. He found it much more 

 difficult to apply the method to such an affirmative question than to 

 negative theorems (cf . Fermat, 2 etc., Ch. XXII) ; for the former, " the 

 method had to be supplemented by some new principles." 



11 Oeuvres de Descartes, II, 92; letter to Mersenne, March 31, 1638. Cf. p. 195. 



12 Oeuvres, II, 203^; letter to Roberval, Aug., 1640. 



13 Memorie Accad. Pont. Nuovi Lincei, 8, 1892, 84-108; Oeuvres de Fermat, 4, 1912, 205-7. 



14 Oeuvres, II, 221; letter to Frenicle, June 15, 1641. 



16 Oeuvres, II, 313, 403; III, 315; letters to Pascal, Sept. 25, 1654, and to Digby, June 19, 1658. 

 18 Oeuvres, II, 432; letter to Carcavi, communicated to Huygens, Aug. 14, 1659. 



