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



In the s l ame way, Nn" (n" < n'} is the sum of four squares; etc., finally 

 N 1 is a sum of four squares. 



He proved that, if N is a prime not dividing the given integers X, ju, v, 

 we can find integers x, y, z not divisible by N such that s = Xz 2 + ju?/ 2 + vz* 

 is divisible by N. Since X is prime to N, we can determine integers m 

 and n such that Xw = n, \n = v (mod N). Then s = is equivalent 

 to a = mb + nc (mod N) for quadratic residues a, 6, c. If the latter is 

 impossible, then mb + n is a non-residue for each of the (N l)/2 residues 

 6 and hence gives all the non-residues. Then if d is any residue, bd is a 

 residue e, so that we + dn must be a non-residue. This exceeds the non- 

 residue me + n by n(d 1) = . For d ^ 1, co is prime to N. Thus, 

 if a is any non-residue, a + co is a non-residue. But a, a + co, , 

 a + (AT l)w are congruent to 0, 1, -, JV 1 in some order and hence 

 are not all non-residues. 



Euler 11 gave a slight modification of his preceding proof. We may 

 assume that p, q, r, s in Nn = p 2 + ( + r 2 + s 2 are numerically < $N t 

 where N Is & prime. Then n < N and we can find integers a, a, - , d } d, 

 such that 



p Na + na, q = Nb + np, r Nc + 717, s = Nd + nd, 



where a, 6, c, d are numerically < \n. Then A^n = N 2 cr + 2JVriA + nH, 

 so that <r = nn'j n' < n. Multiplying by n'/n, we get 



Nn' = (Nn' + A) 2 + B 2 + C 2 + Z> 2 . 

 Euler 12 noted that a 2 + 6 2 + c 2 = 4(z 2 + 3?/ 2 ) = H for 



a = 2m(ps + qr) + 2w(3gs pr), 

 6, c = w{(3g p)s + (9 ^F p)r} + w{3(g =F p)s (3g db p)r}. 



Euler 13 remarked that the sum of two primes of the form 4n + 1 is a SI 

 since each is a 12, and verified that every number 4& + 2 ^110 is a sum 

 of two primes 4n + 1. 



A; M. Legendre 14 remarked that a proof of Fermat's assertion that 

 every prime 8n 1 is of the form p 2 + q 2 + 2r 2 would complete the proof 

 that every number is a SL For, any prime 8n 3 is of form p 2 + <? 2 , 

 any prime 8n + 3 is of form p 2 + 2<? 2 , any prime Sn + 1 is simultaneously 

 of the last two forms. 



Legendre 15 reproduced Euler's 10 proof, using in place of Theorem I its 

 generalization by Lagrange. 



C. F. Gauss 16 subtracted from the given number 4n + 2 any square 

 less than it, from 4n + 1 an even square, from 4n + 3 an odd square. The 

 remainder is = 1, 2, 5 or 6 (mod 8) and hence is a sum of 3 squares. Thus 



11 Opera postuma, 1, 1862, 197-8 (about 1773). 



12 Novi Comm. Acad. Petrop., 18, 1773, 171; Comm. Arith., I, 515. 



13 Acta Acad. Petrop., 4, II, 1780 (1775), 38; Comm. Arith., II, 134-9. 



14 M<5m. Acad. Roy. Sc. Paris, 1785, 514 Cf. Pollock 47 ; also Euler, 12 Lebesgue 31 of Ch. VII. 

 16 Essai sur la thSorie des nombres, Paris, 1798, 198; ed. 2, 1808, 182; ed. 3, 1830, I, 211-6, 



NOB. 151-4 (Maser, I, pp. 212-6). 

 16 Disq. Arith., 1801, art. 293; Werke, I, 1863, 348. 



