300 HlSTOKY OF THE THEORY OF NUMBERS. [CHAP. VIII 



F. J. Studnicka 90 noted that Euler's (1) includes the formula of Cauchy, 22 

 and deduced the like formula expressing a product of three sums of 3 

 squares as a 00 . 



L. Gegenbauer 91 proved new expressions of Jacobi's theorems. The 

 number of representations of an odd number n as a SI equals 8 times the 

 number of divisors of the various g.c.d.'s of n with the numbers ^ n; also 

 equals 8 tunes the sum of the products obtained by multiplying the number 

 of divisors of every factor of n by the number of integers not exceeding the 

 complementary factor and relatively prime to it. The number of proper 

 representations of an odd number n as a SO equals 8 tunes the number of 

 decompositions, into two relatively prime factors, of the various g.c.d.'s of n 

 with the integers ^ n; also equals 8 times the sum of the products obtained 

 by multiplying the number of decompositions of every divisor of n into two 

 relatively prune factors by the number of integers relatively prime to and 

 not exceeding the complementary divisor. 



B. Sollertinski 92 noted [Catalan 66 ]] that a CS is a ffl : 



P* = m 2 -f- 



E. N. Barisien 93 noted that s 5 is a GO if s = x 2 + 2/ 2 , since 



S 2 = (3.2 _ ^2)2 + 4 X Y, s 3 = (3xy* - z 3 ) 2 + (3x*y - i/ 3 ) 2 . 



[We may conclude that s 5 is a (21, not merely a SI.] 



G. Wertheim 94 proved that every prime p divides a S3 as had Matrot, 84 

 and also by finding how often in the series 1, 2, , p 1 a residue follows 

 a residue, or a quadratic non-residue follows a residue. 



L. E. Dickson 95 exhibited all solutions of x 2 + 2/ 2 = 1 (mod p} and of 

 x 2 + 2/ 2 = (mod5 4 ). 



K. Petr 96 proved two formulas by Gauss (Werke, III, 476) on theta 

 functions by the method outlined by Gauss. From them are derived 

 relations giving the number <p(N), t(N), t'(N) of representations of N by 



x 2 + i/ 2 + 92 2 + 9w 2 , x 2 + 2/ 2 + z 2 + 9w 2 , x z + 9^/ 2 + 9z 2 + 9w 2 , 

 respectively. Let x(N) be the known number for four squares. Then 

 *>(#) = i(xW + 162(- l) 3z+y)/6] z}, A^ + (mod 3), 



summed for all positive odd solutions of 3x- + y 2 = 4N. For N divisible 

 by an odd power of 3, <f>(N) = 0; if by an even power of 3, <p(N) = 

 Also, 



0> -/V ^ (mod 3) 



' ^ ; 



90 Prag Sitzungsber. (Math. Naturw.), 1894, XV. 



91 Sitzungeber. Akad. Wiss. Wien (Math.), 103, Ha, 1894, 121. 



92 El Progreso Matemdtico, 4, 1894, 237. 



93 Le matematiche pure ed applicate, 1, 1901, 182-3. 



94 Anfangsgfunde der Zahlenlehre, Braunschweig, 1902, 396. 

 96 Amer. Math. Monthly, 11, 1904, 175; 18, 1911, 43-4, 118. 

 98 Prag Sitzungsber. (Math. Naturw.), 1904, No. 37, 6 pp. 



