318 HISTORY OF THE THEORY OF NUMBERS. [CHAP. IX 



R. Goormaghtigh 56 proved that every power of an even [odd] integer 

 with an exponent =s 3 is a sum of 5 [6] squares > 0. If n is odd and > 1 

 and if a > 0, n 4a+l is a sum of 5 squares > 0. 



Mordell 57 employed the theory of modular functions to find the number 

 of representations as a sum of 2r squares. 



G. H. Hardy 58 deduced from the theory of elliptic functions the number 

 of representations as a sum of 5 or 7 squares. This investigation, con- 

 tinued by S. Ramanujan, 58a led to a complete solution of the problem of 

 the representation of a number as a sum of n squares for n < 8, and to 

 asymptotic formulas for any n. The method used is an application of. the 

 general theory cited in Ch. III. 221 



E. T. Bell 586 proved Liouville's 10 - n formulas by use of series for elliptic 

 functions and stated that they are only the first cases of an infinitude of 

 similar results which may be found by using higher powers than the first 

 and second, or products, of the series. 



On 10 odd squares, see Pollock 117 of Ch. I. On 8 squares, see Sier- 

 pinski 158 of Ch. VI. For 5 squares, see Hermite 69 and Humbert 108 of Ch. 

 VII. In Ch. XI are noted Liouville's results on sums of n squares for 

 n = 8 and 12 and in papers 6 and 7 minor results f or n = 5 and 7. 



RELATIONS BETWEEN SQUARES. 

 The Japanese Aida Ammei 59 proved between 1807 and 1817 that 



Xi = - a\ + a\ -f a\ + - - + a 2 n , x r = 2i(z r (r = 2, -, ri), 



satisfy x\ + ---- h a = 2/ 2 . This result was known to Euler 191 - 294 > 308 of Ch. 

 XXII. Ajima Chokuyen, 59 " in a manuscript dated 1791, had solved 

 x\ -\- - + xl = y 2 in integers. 



It was proved by J. R. Young, 60 who proved also the identity 



+ Zfayj - Xjy^) 2 (i, j = 1, , n; i < j}. 



The latter was proved otherwise by A. Cauchy. 61 



Aida's result has been published also by D. S. Hart and A. Martin, 62 

 E. Catalan, 63 A. Martin, 64 and G. Bisconcini 65 (by geometrical considerations 



66 L'interme'diaire des math., 23, 1916, 152-3. 



57 Quar. Jour. Math., 48, 1917, 93-104. 



68 Proc. Nat. Acad. Sc., 4, 1918, 189-193. Proc. London Math. Soc., Records of Meeting, 



March, 14, 1918. 



680 Trans. Cambr. Phil. Soc., 22, 1918, 259-276. 

 686 Bull. Amer. Math. Soc., 26, 1919, 19-25. 

 68 Y. Mikami, Abh. Gesch. Math. Wiss., 30, 1912, 247. Based on C. Hitomi's article in 



Jour. Phys. School of Tokyo, 15, 1906, 359-62. 

 690 Jour. Phys. School of Tokyo, 22, 1913, 51. 



60 Trans. Roy. Irish. Acad., 21, II, 1848, 333. 



61 Cours d'analyse de l'6cole polyt., 1, 1821, 455-7. 



62 Math. Quest. Educ. Times, 20, 1874, 83; 63, 1895, 49, 112. 



63 Bull. Acad. Roy. Sc. Belgique, (3), 27, 1894, 10-15. 



94 Proc. Edinburgh Math. Soc., 14, 1896, 113-5; Math. Mag., 2, 1898, 209. 

 86 Periodico di Mat., 22, 1907, 28. 



