CHAP. VI] SUM OF TWO SQUARES. 229 



Frenicle 17 concluded from numerical tables that, if. p if p 2 , are 

 distinct primes, each the hypotenuse of a right triangle (a necessary and 

 sufficient condition being that the prime is of the form 4k + 1), a number 

 N = p\ l . . p" n n is the hypotenuse of exactly 2 n ~ l primitive right triangles 

 (i. e., with relatively prune legs). He recognized that the problem reduces 

 to the question of the number of ways in which the proposed number N 

 can be expressed as the product of two relatively prime factors. The non- 

 primitive triangles are obtained from the primitive triangles whose hypot- 

 enuses are the factors of N. Fermat's rule D is given. Problem G is dis- 

 cussed (pp. 34-46). 



John Kersey, 18 to treat x 2 + y 2 = d? + 6 2 of Diophantus II, 10, set 

 x = ra + b, y = sa d. Thus a = 2(sd rb)/(s 2 + r 2 ), so that the values 

 of x, y follow. He also treated the problem pBachet, 7 304] with the restric- 

 tion that x or y shall fall within given limits. 



Claude Jaquemet, 19 in a letter Jan. 26, 1690, proved that an integer not 

 a square, which divides no sum of two squares without dividing each 

 square, is not a sum of two squares, integral or fractional. A manuscript 

 by Jaquemet or N. Malebranche proved also that a number which 

 divides a sum of two relatively prime squares is itself a sum of two squares; 

 but the later proof by Euler 24 is far simpler. Cf . Bhdscara, 30 88, of Ch. XII. 



The Japanese Matsunago, 20 the first half of the 18th century, would 

 solve x 2 + y 2 = k by setting k/2 = r* + R, where r 2 is the greatest square 

 contained in k, and forming the equations 



i = 2r 1, a 2 = ai 2, a 3 = a 2 2, , 

 &! = 2r + 1, 6 2 = &! + 2, 6 3 = 62 + 2, 



From 2R subtract successively 61, b z , When a difference is negative, 

 add the corresponding a*. If the remainder zero is reached, and a', b f 

 are the values last employed, a solution is 



x = W + 1), y = |(6' - 1). 

 It was stated that a set of solutions of x 2 + y z = z 3 is given by 



x = (ra 2 3n 2 )m, y = (3m 2 n 2 )n, z = m 2 + n 2 . 



L. Euler 21 proved that, if neither a nor b is divisible by the prime 

 p = 4n 1, then a 2 + b 2 is not divisible by p. For, a 4 "" 2 6 4n ~ 2 is 

 divisible by p and hence a 4n ~ 2 + & 47l ~ 2 is not; thus the factor a 2 + 6 2 of 

 the latter is not divisible by p. 



Euler 22 stated that if 4m + 1 is composite it is either not a sum ED of 

 two squares or is so in more than one way; if ab and a are E], 6 is a El. 



17 M6m. Acad. Roy. Sc., 5, 1666-99, ed. Paris, 1729, 22-34, 156-163. 



18 The Elements of Algebra, London, Book 3, 1674, 9-17, 20-23. 



19 Bull. Bibl. Storia Sc. Mat. e Fis., 12, 1879, 890^, 644; 13, 1880, 444. 



20 Y. Mikami, Abh. Geschichte Math. Wiss., 30, 1912, 233. 



21 Correspondence Math. Phys. (ed., Fuss), 1, 1843, 117; letter to Goldbach, March 6, 1742. 



Novi Comm. Acad. Petrop, 1, 1747-8, 20; Comm. Arith., I, 53, 16. French transl. 

 in Nouv. Ann. Math., 12, 1853, 46. 

 ^Corresp. Math. Phys. (ed., Fuss), 1, 1843, 134, letter to Goldbach, June 30, 1742. 



