. Carnegie Institution of Washington publication. 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"AT, where N is one of the preceding three types. Gauss17 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 Hermite69] (6) (l = (1 - 2y + 2y4 ---- )4 + (2
. Carnegie Institution of Washington publication. 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"AT, where N is one of the preceding three types. Gauss17 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 Hermite69] (6) (l = (1 - 2y + 2y4 ---- )4 + (2?/1/4 + 2t/9/4 He noted (p. 445) that [cf. Legendre,23 Jacobi,24 and Genocchi39] (7) (l +. Gauss18 noted that every decomposition of a multiple of a prune p into a2 + 62 + c2 + d2 corresponds to a solution of z2 + yz + 22 = 0 (mod p) proportional to a2 + fe2, 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 + xz + ?/2 = 0 (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? + 62; 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. Malfatti19 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 Knz = p2 + g2, 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 = a2 + 62, 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, Xn2 = p2 + qz + r2, 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.
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