CHAPTER VI. 



SUM OF TWO SQUARES. 



Diophantus, II, 10, divided a given number 13 = 2 2 + 3 2 , which is a 

 sum of two squares, into two other squares, (z + 2) 2 + (mz 3) 2 , by taking 

 m = 2, whence z = 8/5. In III, 22, Diophantus required four numbers x 

 such that each of the eight expressions E = (So;;) 2 x t shall be a square. 

 In any right triangle (p, b, h), h 2 2pb = D. [If h 2 = pi + b\ (i = 1, 

 -, 4), take z t - = 2p i b i x 2 , Zx* = hx; then E = x 2 (h 2 2pi6<) = D.j 

 Hence we seek four right triangles with equal hypotenuses. We must 

 therefore find a square which can be expressed as a sum of two squares in 

 four ways. Take the right triangles (3, 4, 5) and (5, 12, 13) ; multiply the 

 sides of each by the hypotenuse of the other. We obtain the triangles 

 (39, 52, 65) and (25, 60, 65) with equal hypotenuses. The number 65 can 

 be expressed as a sum of two squares in two ways: 65 = 4 2 + 7 2 = I 2 + 8 2 , 

 since 65 is the product of 13 and 5, each a sum of two squares. Now form* 

 the right triangle (33, 56, 65) from 7, 4 and (16, 63, 65) from 8, 1. We now 

 have four right triangles with equal hypotenuses. [If we carry out the 

 corresponding process on the right triangles (a 2 b 2 , 2ab, a? + 6 2 ), (c 2 d 2 , 

 2cd, c 2 + d 2 ), we obtain by multiplication two triangles with the hypotenusef 



(1) (a 2 + 6 2 )(c 2 + d 2 ) = (ac bd) 2 + (ad =F be) 2 . 



The right triangles formed from ac bd and ad T be give two new tri- 

 angles with the same hypotenuse, provided c/d is distinct from a/6, b/a, 

 (a d= 6)/(o T &).] 



Diophantus, V, 12, treated the division of unity into two parts such 

 that, if a given number a is added to each part, the sums are (rational) 

 squares. The problem is equivalent to the representation of 2a + 1 as a 

 sum of two squares. It is stated that a must not be odd [so that no number 

 4n 1 is a sum of two squares]. Unfortunately the text of the second 

 part of the necessary condition is very obscure. C. G. J. Jacobi 1 emended 

 it to read that 2a + 1 must have no factor of the form 4n 1 ; P. Tannery 

 and T. L. Heath, in their editions of Diophantus, read prime factor; but 

 neither correction makes the criterion exact. 



Diophantus, VI, 15, stated that 15 is not a sum of two (rational) squares. 



Mohammed Ben Alhocain, 2 an Arab of the tenth century, gave a table 

 of numbers equal to a sum of two squares, formed by adding each square to 

 itself and to the larger squares. It is stated falsely that if an even number 

 is a sum of two squares, one of them is unity. 



* See Ch. IV, Diophantus. 7 



t For a like composition of factors a 2 eb z , see Euler 66 of Ch. XII. 



^erichte Akad. Wiss. Berlin, 1847, 265-278; Werke, 7, 1891, 332-344 (report below). 



Same by H. Hankel, Zur Geschichte der Math., 1874, 169. 

 2 Cf. F. Woepcke, Atti Accad. Pont. Nuovi Lincei, 14, 1860-1, 306-9. 



16 225 



