292 HISTORY OF THE THEORY OF NUMBERS. [CHAP, vm 



J. Liouville 490 proved that the number of representations byz 2 +?/ 2 +z 2 +4Z 2 

 of an odd number m is {4 + 2( l) (m ~ 1)/2 }o-(m), of 2m is 12o-(m), of 4m is 

 8cr(?ri), of 2 a m(a^3) is 24<r(m). The number of representations by 

 x 2 + 4?/ 2 + 4z 2 + 4Z 2 of m = 41 + 3 is zero, of m = 4Z + 1 is 2<r(m), of 2m is 

 zero, of 4m is 8<r(m), of 2 a m (a ^ 3) is 24<r(m). He found also the number 

 of proper representations by these forms. He 496 expressed the number of 

 representations of 2 a m by x* + ay* + bz 2 + 16J 2 for (a, 6) = (4, 4), (16, 16), 

 (4, 16), (1, 16), (1, 4), (1, 1), in terms of a(m) and 2(- l) (i - 1)/2 i, summed 

 for the odd integers i for which m = i 2 + 4s 2 . From Jacobi's 24 result, he 49c 

 derived also the number of representations by x 2 + y 2 + 9z 2 + 9 2 . 



J. Liouville 50 considered an odd integer m and the decompositions 



4m = i 2 + i; + i\ + il, 2m = r 2 + r\ + 4s 2 + 4sJ, 

 where i, ii, i* 2 , 13, r, ri are positive odd integers, and stated that 



J. Plana 51 proved Jacobi's 24 formula 



(1 + 2q + 2q* + 2 ? 9 + . -) 4 = 1 + 



H. J. S. Smith 52 discussed Jacobi's 24 - 25 theorems that the number of 

 representations of an odd number m as a SO is 8<r(m); the number of 

 representations of 4m as a sum of four odd squares is 16o-(m). 



F. Pollock 53 stated that the algebraic sum of the roots in some repre- 

 sentation of a given odd number as a 3] will equal any assigned odd number 

 not exceeding the maximum; that the difference of some two of the roots 

 will equal any number not exceeding the maximum. But all that is defi- 

 nitely proved in this paper, dealing with numerical statements, is that 

 any number n is a sum of four triangular numbers, since Bachet's theorem 

 gives 



4rc + 2 = (2a + I) 2 + (26 + I) 2 + (2c) 2 + (2d) 2 , 



n = (a 2 + a + c 2 ) + (6 2 + 6 + d 2 ). 



V. Bouniakowsky 54 employed the known result that the quadratic 

 residues of a prime p = 4n + 1 may be paired so that the sum of a pair is p, 

 and likewise the non-residues, to obtain relations like 



10 2 + II 2 = 2 2 + 3 2 + 8 2 + 12 2 , 6 2 + 7 2 = I 2 + 2 2 + 4 2 + 8 2 (p = 17), 

 13 3 = I 3 + 5 3 + 7 3 + 12 3 , 13 3 + 14 3 = I 3 + 3 3 + 17 3 



[the first from 2 2 + 3 2 = 13, 8 2 + 12 2 = - 1 + 1 (mod 13)]. 



490 Jour, de Math., (2), 6, 1861, 440-8. Cf. Liouville 2 of Ch. XI. 

 *' b lbid., (2), 7, 1862, 73-6,77-80, 105-8, 117-20, 157-60, 165-8. 

 tte lbid., (2), 10, 1865, 14-24. 



60 Jour, de Math., (2), 8, 1863, 431-2. 



61 Mem. Accad. Turin, (2), 20, 1863, 130. 



62 British Assoc. Report, 1865, 337; Coll. Math. Papers, I, 307. 



63 Proc. Roy. Soc. London, 15, 1867, 115-127; 16, 1868, 251-4; abstract of Phil. Trans., 158, 



1868, 627-642. His " proof " of Bachet's theorem is given in Ch. I. 124 

 M Bull. Acad. Sc. St. Peterebourg, 13, 1869, 25-31. 



