CHAP. VIII] SUM OF FOUR SQUARES. 283 



the given number is a sum of 4 squares. Finally, a multiple of 4 is of the 

 form 4"A T , where N is one of the preceding three types. 



Gauss 17 noted that the theorem (1) that a product of two sums of four 

 squares is a SI is represented in the simplest way by 



(Nl + Nrri)(N\ + Nrf = N(l\ + wi/i) + N(W - mX'), 



where N denotes the norm and I, m, X, ju, X', // are complex numbers, 

 X, X' and /*, /*' being conjugate imaginaries. He noted (p. 447) that [cf. 

 Glaisher, 59 Hermite 69 ] 



(6) (l 



= (1 - 2y + 2y 4 ---- ) 4 + (2?/ 1 / 4 + 2t/ 9 / 4 



He noted (p. 445) that [cf. Legendre, 23 Jacobi, 24 and Genocchi 39 ] 



(7) (l + 



Gauss 18 noted that every decomposition of a multiple of a prune p 

 into a 2 + 6 2 + c 2 + d 2 corresponds to a solution of z 2 + y z + 2 2 = (mod p) 

 proportional to a 2 + fe 2 , ac + bd, ad be or to the sets derived by inter- 

 changing 6 and c or 6 and d. For p = 3 (mod 4), the solutions of 

 1 + x z + ?/ 2 = (mod p) coincide with those of 1 + (z + ^2/) p+1 = 0. 

 From one value of x + iy we get all by using 



(x + %)(w + i)/(w - i) (u = 0, 1, -, p - 1). 



For p = 1 (mod 4), p = a? + 6 2 ; then 6(w + i)/{a(u i)} give all values 

 of x + % if we exclude the values a/b and b/a of w. 



G. F. Malfatti 19 did not prove as he promised to do that every integer 

 is a (2 . After verifying this for about 50 small numbers, he considered the 

 equation Kn z = p 2 + g 2 , where K is a given integer. If we admit his 

 assertion that K must be a El, the equation has evident solutions with 

 n = 1. Taking K = a 2 + 6 2 , he found an infinitude of solutions, with / 

 and g arbitrary, by setting 



an q p bn \ 



ff) = 



, 

 The equation obtained by eliminating p is satisfied if we take 



Next, Xn 2 = p 2 + q z + r 2 , in which we may limit K to be odd or the double 

 of an odd number, and n to be odd, is said without adequate proof to be 



17 Posth. MS., Werke, 3, 1876, 383-4. 



18 Posth. paper, Werke, 8, 1900, 3. 



19 Memorie di Mat. e Fis. Soc. Italiana Sc., Modena, (1), 12, pt. 1, 1805, 296-317. 



