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



and get the identity 2a*-2Z; = STT^, where 



etc. By permuting the a's or changing the signs, we get 96 formulas (1). 



E. Catalan 80 made an invalid criticism of Legendre's 15 proof that every 

 prime is a SI, who is said to have assumed that every integer N has a prime 

 divisor > V^T. Catalan's remark (p. 164) that if N and A are sums of 

 four integral squares, their quotient N/A is a sum of fractional squares, 

 was known to Euler. 8 Catalan proved that every integer is a sum of four 

 fractional squares in an infinitude of ways and stated (p. 212) that every 

 number Sn + 4 is a sum of four odd squares of which two are equal. 



M. A. Stern 81 gave an elementary proof of Jacobi's 24 theorem. Let m 

 be odd. The number of representations of 2m as a S3 is three times that 

 of m, since m = p 2 + q 2 + r 2 + s 2 implies 



2m = 2(p q} 2 + 2(r d= s) 2 = 2(p r) 2 + 2(q s) 2 



= 2(p dr s) 2 + 2(g r) 2 . 



Conversely, if 2m = a 2 + /3 2 + 7* + 5 2 , two of the squares are even and 

 two odd, so that 



m = 



Repeating the process, we get [cf. Zehfuss 46 ] 



4m = (2p) 2 + (2q) 2 + (2r) 2 + (2s) 2 , 



(14) 4m = (p + q + r dh s) 2 + (p + q - r =F s) 2 



g + rTs) 2 + (p-?-r s) 



Conversely, 4m = Za 2 implies 2m = {^( + 0)} 2 +'. Hence 4m and 

 2m have the same number of representations as a SI. It is shown that if 

 2 a m and 2 a+1 m have the same number v of representations, then 2 l m(t ^ a) 

 has v representations. If m = 4/c + 1, three of the numbers p, g, r, s 

 are even and the fourth is odd, so that the squares in (14) are all odd. If 

 m = 4k + 3, three of p, q, r, s are odd and one is even, and the preceding 

 conclusion holds. By Jacobi's 30 theorem, there are 16<r(m) representations 

 of 4m by four odd squares. Hence if pqrs =j= 0, there are 8<r(w) representa- 

 tions of 4m by four even squares and hence 24o-(m) representations in all. 

 This result is proved to hold also if pqrs = 0. Cf. Vahlen. 88 



T. Pepin 82 proved Jacobi's 25 theorem that, if m is odd, the number of 

 decompositions of 4m as a sum of 4 odd squares with positive roots is 

 o-(m), by taking t = ir/2 in a formula involving sums of shies of multiples 

 of t. The number of representations of 2m by z 2 + y 2 + 4s 2 + 4Z 2 is 4<r(m). 

 The number of representations of 2m by z 2 + y 2 + z 2 + P or z 2 + y 2 + z 2 

 with x + y = 1 (mod 2), is 16<r(m) or 80- (m) respectively. He gave 



80 M<5m. Soc. Roy. Sc. de Li&ge, (2), 15, 1888, 160 (Melanges Math., III). 



81 Jour, fiir Math., 105, 1889, 251-262. 



82 Jour, de Math., (4), 6, 1890, 19-20. 



