CHAP. VIII] SUM OF FOUR SQUARES. 295 



factor in M(l + l/a)(l + 1/6) ways. (V) If r l} pi, p 2 are the numbers 

 of sets of distinct integral solutions not zero of 



x 2 + y 2 + z 2 + t 2 = 2(2n - 1), x 2 + y 2 + z 2 + t 2 = 2n - 1, 



2x 2 + y 2 + z 2 = 2n - 1, 



then T! = 3pi + p 2 . (VII) If re 2 + ?/ + z 2 + t 2 = 4(2n - 1) has s t sets of 

 distinct odd integral solutions and x 2 + y 2 + z 2 = 2n 1 has p 4 sets of 

 distinct solutions =t= 0, then s x = 2pi + p 4 . (IX) If S 4 denotes x 2 + 2/ 2 

 + z 2 + 2 , the number of sets of solutions of 2 4 = 2*(2n 1) is expressed 

 in terms of the numbers of sets of solutions of S 4 = 2n 1 and Z 3 = 2n 1 

 and the number of sets of solutions when two or three variables are equal. 



E. Fergola 65 had stated the preceding theorem (V), and (I) with the 

 restriction that 2n 1 is not a square. 



E. Catalan 66 noted that 2p = a + 6 + c implies 



p 2 + (p - a) 2 + (p - 6) 2 + (p - c) 2 = a 2 + b 2 + c 2 



and gave various identities in a, b, c, which express the square of the sum 

 of three squares as a ID . 



J. J. Sylvester 67 proved that any prime p is a divisor of x 2 + y 2 -f- 1. 

 Assume the contrary. Then p 4= 4i + 1 since p does not divide x 2 -f 1. 

 Let p be any primitive pih root of unity and set R = Sp* 2 , summed for 

 the quadratic residues x 2 < p. Let R' be the period conjugate to R. 

 Expand R 2 as a sum of powers of p. Since p =1= 4i -f 1, x 2 + 2/ 2 =4= p and 

 no pth power of p can occur in the expansion of R 2 . Since, by hypothesis, 

 neither 2x 2 nor x 2 + y 2 is = 1 (mod p), no such power as p p-1 can appear 

 in R 2 , while it belongs to R'. Thus no term of R' appears in R 2 . As each 

 power of p in R 2 belonging to the same period must appear a like number 

 of times, we have R 2 = R(p l)/2, whereas R =t= or (p l)/2. 



From this theorem follows Bachet's theorem. A similar proof shows 

 that Ax 2 + By 2 + Cz 2 = (mod p) is solvable. 



H. J. S. Smith 68 indicated a proof of Bachet's theorem by continued 

 fractions. 



C. Hermite 69 proved (6) by elliptic functions and concluded that the 

 number of decompositions into four squares of any odd integer n equals 

 8 times the number of decompositions of 4n as a sum of four squares whose 

 roots are odd and positive. Cf . Jacobi. 25 



J. W. L. Glaisher 70 considered the cr(N) compositions (allowing permuta- 

 tions) of 4JV as a sum of 4 odd squares, took the square root of the first 

 square (for example) in each such composition, giving it the sign accord- 

 ing as it is of the form 4m =b 1, and formed the algebraic sum A of these 

 square roots. Next, consider the compositions of 2N as a sum of 2 odd 

 squares, take the product of the square roots of the two squares in each such 



86 Giornale di Mat., 10, 1872, 54. 



66 Nouv. Corresp. Math., 5, 1879, 92-93. 



67 Amer. Jour. Math., 3, 1880, 390-2; Coll. Math. Papers, 3, 1909, 446-8. 



88 Coll. Math, in memoriam D. Chelini, Milan, 1881, 117; Coll. Math. Papers, II, 309. 

 89 Cours, Fac. Sc. Paris, 1882; 1883, 175; ed. 4, 1891, 242. 

 "Quar. Jour. Math., 19, 1883, 212-5; 36, 1905, 342-3. 



