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



Hence 



I = (2p + I) 2 + (2q + I) 2 + 4r 2 + 4s 2 , 



= (P + q + I) 2 + (P ~ tf + (r + *) 2 +(r~ s) 2 . 



As a corollary, 2 A = SI implies A = SI . 



On March 24, 1750 (p. 513), Goldbach had stated that there is a definite 

 connection between the sets of four squares whose sums are 2m 1 and 

 2m + 1, as derived from 8m + 3 = SI. June 9, 1750 (p. 518), Euler 

 interpreted this as follows: From 8m 5 = a 2 + fc 2 + c 2 , where a, b, c 

 are odd, 



where two of the squares are even. Set 2p = (a + l)/2, 2g = (b + c)/2. 

 Then 



4m - 2 = (2p) 2 + (2g) 2 + r 2 + s 2 , 



2m - I = (p + <?) 2 + (p - 

 /a6cdb 



lV 

 /' 



2 



where two or four signs are +. From 8m + 4 = 9 + a 2 + b z -\- c 2 , 



/a + SV -3V 6 + cV /6 - cV 



J 



/a6 d=c3V 

 2w+l = 2J^- -y J , 



where two or four signs are + . Hence, from 8m 5 = S] , 

 2m - 1 = p z + q* + r 2 + s 2 , 

 2m + 1 = (p + I) 2 + (q + I) 2 + (r - I) 2 + (s - I) 2 . 



Thus r-fs p q = 1 and we can express any odd number as a sum of 

 four squares the algebraic sum of whose roots is unity. [CL Cauchy, 1813]. 

 Euler stated (p. 521, p. 527, and again on Dec. 4, 1751, p. 559) that while 

 he had proved that any rational number is the sum of four rational squares, 

 he had not proved the theorem for integral squares. 



Goldbach (p. 526) noted that a, ft, % + + 7 + 25, and a. + + d, 

 a + 7 + 5, $ + 7 + 5, 5, and a + 8, |8 + 5, 7 + 5, a + P + v + 8 have 

 the same sum of squares. 



Euler, July 3, 1751, p. 542, discussed the problem to make 



S = 2 _|_ p _|_ y + g2 + e 



a ffl . Call the roots a kx, ft mx, 7 nx, d + x. Then 



6 = A - |z + ~ , A = &a + m/3 + 717, 5 = k* + m 2 + n z + 1. 



