262 HISTORY OF THE THEORY OF NUMBERS. [CHAP, vii 



C. F. Gauss 20 determined the number <f>(m) of proper representations 

 x, y, z, without common factor (and counted as different from y, x, z and 

 from x, y, z\ etc.) of an integer m as a G2 . Let h be the number of classes, 

 in the principal genus, of the properly primitive binary quadratic forms of 

 determinant m. Let /* be the number of distinct prime factors of m. 

 Then 



<f>(m) = 3-2 +z h if m = 1, 2, 5, 6 (mod 8), 



0(m) = 2 +z h if m = 3 (mod 8). 



In particular, we have Legendre's 19 theorem. But the squares of x, y, z\ 

 x, y, z] y, x, z', etc. give the same decomposition of m into a O. The 

 resulting number of decompositions (art. 292) of m agrees with that derived 

 by (incomplete) induction by Legendre 16 for the case m a prime. 



A. Cauchy 21 noted, as a corollary to Legendre's theorem, 19 that if a is 

 any integer and if 4" is the highest power of 4 dividing a, then o is a S if 

 and only if o/4 a is not of the form Sn + 7. 



J. R. Young 210 solved a; 2 + y~ + z 2 = w 2 by taking w = x + P and 

 finding x rationally, or by setting y 2 = 2xz. Then if w is given, take y = pz, 

 whence z is found in terms of p. To find (p. 346) three numbers in har- 

 monical progression whose sum of squares is a square, take l/(x y), 1/x 

 as the three numbers; the condition 3# 4 + y 4 = D is satisfied if x = 2, 



y-l. 



C. Gill 216 noted that the sum of the squares of 2mn(k* + I 2 ), 2kl(m 2 - n 2 ) 

 and (k z - Z 2 )(m 2 - n 2 ) equals the square of (k* + Z 2 )(m 2 + n-}. 

 C. G. J. Jacobi 22 proved by use of elliptic functions that 



(1) { Z (- I) m a; (3m2+w)/2 I* = Z (- l) n (2n + 



I m= oo J n=0 



a result occurring also in Gauss' posthumous papers. 



Jacobi 23 gave an elementary proof of (1). Replace x by x 24 and multiply 

 the resulting equation by z 3 ; we get 



(2) I Z (~ il)^^ +1 > 2 1' = Z (- 1) (6 - 1} V (6 odd, 6 > 0). 



I m= oo ) 6 



For m positive, set a = 6m + 1 ; for m negative, set a = 6m 1 ; thus 



(E * a2 ) 3 = 



where a and b range over all positive odd integers such that a is not divisible 

 by 3. The sign in the left member is + if a = 12k d= 1, if a = '12k 5. 

 The expansion gives the following theorem: If a number 24k -\- 3, not of 

 the form 36 2 , be expressed as a sum of three squares (6m I) 2 in all possible 



20 Disq. Arith., 1801, Art. 291; Werke, 1, 1863, 343; German transl. by H. Maser, pp. 329-33. 



Cf. H. J. S. Smith, British Assoc. Report, 1865; Coll. Math. Papers, I, 324. 



21 Me"m. Sc. Math. Phys. de 1'Institut de France, (1), 14, 1813-5, 177; Oeuvres, (2), VI, 323. 

 21a Algebra, 1816; S. Ward's Amer. ed., 1832, 326-7. 



216 The Gentleman's Math. Companion, London, 5, No. 29, 1826, 364. 



22 Fund, nova func. ellip., 1829, 66(7); Werke, I, p. 237 (7). 



23 Jour, fur Math., 21, 1840, 13-32; Werke, VI, 281-302. French transl., Jour, de Math., 



7, 1842, 85-109. 



