CHAP. VIII] SUM OF FOUR SQUARES. 277 



then 2 (4m + 2) is a 030 and hence 2m + 1 is a SL May 4, 1748 (p. 452), 

 he gave the fundamental formula (Cf . Euler 165 of Ch. XIX) 



[(a 2 + 6 2 + c 2 + d 2 )(p 2 + q 2 + r 2 + s 2 ) = x 2 + y 2 + z 2 + v 2 , 

 (1) i x = ap + bq + cr + ds, y = aq bp cs =F dr, 

 [ 2 = ar =F 6s cp dg, v = as 6r =F cq dp, 



and stated (p. 454, and Aug. 17, 1750, p. 531) that Bachet's theorem would 

 follow if the fourth power of 1 + x + 4 + x g + # 16 + contained x n 

 with a coefficient =t= 0. April 12, 1749 (pp. 495-7), he stated that he had a 

 proof that, if p is any prime, there exist four integers a, , d, each not 

 divisible by p, such that a 2 + + d 2 is divisible by p. Set a = ap x, 

 - , d = Sp =b v, where ^ x ^ %p, - , ^ v ^ ^p. Hence x 2 + + y 2 

 is divisible by p. If p is odd, x < f p, , so that x 2 + + w 2 < p 2 . 

 To prove that every prime is a 30, suppose there is a minimum prune p 

 not a 3D. But z 2 + + y 2 = pq, q < p. Euler believed, but could not 

 prove, that if pq = 30, p =J= 3D, then q 4= SI. Admitting this, we would 

 have a contradiction with the assumption about the minimum p. Thus 

 every prime is a S] and hence by (1) every integer is a SI. 



On the point here left in doubt that pq = 3] and q = ID imply p = SI, 

 Euler proved, July 26, 1749, pp. 505-10, that, if* m ^ 7, wA = SI and 

 m = 3D imply A = SI . Set 



m = a 2 + 6 2 + c 2 + d 2 , 

 = (/ + wp) 2 + (g + rnqY + (h + mr) 2 + (k + ms) z , 



[where /, , k are numerically ^ w/2]. Then / 2 + + & 2 is divisible 

 by m. For m ^ 7, the quotient was verified to be a SI, 



/ 2 + g 2 + /i 2 + k 2 = 

 and pn accord with, but not a consequence of, (1)] 



/ = aX + 67 + cZ + dV, g = bX - aY - dZ + cV, 

 h = cX + dY - aZ -bV, k = dX - cY + bZ - aV, 



A = X 2 + Y 2 + Z 2 + 7 2 + 2(fp + gq + hr + ks) + m(p 2 + q 2 + r 2 + s 2 ) 

 = (x + X) 2 + (y- Y) 2 +(z- Z? + (v - VY 



where x, - -, v are given by (1) with the upper signs. Moreover, he gave 

 a proof of Chr. Goldbach's assertion of June 16 (p. 503) that the sum s of 

 four odd squares can be expressed as a sum of four even squares. Since 



J(2p + I) 2 + 1(2? + I) 2 = (P + q + I) 2 + (p - q} 2 , 

 I = (a + b + I) 2 + (a - 6) 2 + (c + d + I) 2 + (c - d) 2 . 



The last sum involves two even and two odd squares since s = 8m + 4. 



* For the general case Euler 8 admitted in 1751 that he had no proof. 



