CHAP. VIII] SUM OF FOUR SQUARES. 299 



various theorems on the representations of 2 k m by forms 



x 2 + (2 a yY 



E. Catalan 83 noted that, if k = 2a 2 + 3, k is a SO and 7c 2 a SI. 



A. Matrot 84 duplicated in essence the proof by Euler 10 except as regards 

 the theorem that every prime p divides a sum of 2 or 3 squares. Let 

 p = 2h -f- 1. Consider the couples j, 2h j (j = 1, , h 1). If both 

 terms , i of some couple are quadratic residues of p, a = A 2 , a\ = A\, 

 A 2 + A\ + 1 = (mod p). But if no couple is composed of two quadratic 

 residues, the number of residues contained in the couples is ^ h 1. 

 Hence one of the numbers h, 2h, not lying in a couple, is a quadratic residue 

 (there being h such). If h = A 2 , A 2 + A* + 1 = (mod p). If 2h = A 2 , 

 A 2 + 1 = (mod p}. 



E. Humbert 85 proved that if p is odd and =|= 3, 9, at least one of the 

 numbers |(p + 1), ?(p + 3), -, p 1 is a square. Hence if the abso- 

 lutely least quadratic residues of a prime p > 3 be arranged in increasing 

 order of numerical value, the series contains negative terms. Hence if 

 p = 4n + 3, there exsits a positive residue a. followed by the residue a 1. 

 Then a = x 2 , -a - 1 = y 2 , x 2 + y- + 1 = (mod p). 



R. F. Davis 86 noted that, ifs = a + & + c + dis even, a 2 + b 2 + c 2 + d z 

 is expressible as a sum of four new squares by means of the identity of 

 Zehfuss 46 (divided by 4). If s is odd, add m 2 to each member and transform 

 into aSl. R. W. D. Christie made use of various formulas expressing a G3 

 as a SI after proper selection of three of four squares. 



A. Matrot 87 noted that, if p = 2h + 1 is a prime, we can find two 

 consecutive integers a and a + 1 satisfying x h = 1 and x h = 1 (mod p), 

 respectively. For, otherwise 1, 2, , p 1 would all satisfy the first. 

 Hence 



a h+i + ( a + i)w-i -j-i = a _( a + i) + i = o (mod p). 



For p = 3 (mod 4), h + 1 is even, and p divides a ED. His proof that 

 every prime p = 1 (mod 4) divides a E] was quoted under that topic. 



K. Th. Vahlen 88 gave essentially the same argument as had Stern. 81 

 His proof of Bachet's theorem is given in Ch. VII. 74 



E. Catalan 89 gave Legendre's 15 proof of Bachet's theorem. Euler 8 gave 

 the empirical theorem that an integer is not a sum of four fractional squares 

 unless it is a sum of four integral squares. This is said to be false since 

 every integer is a sum of four fractional squares in an infinitude of ways. 



83 Assoc. frang. av. sc., 20, 1891, II, 198. 



84 Assoc. frans. av. sc. (Limognes), 19, 1890, II, 79-81 [20, 1891, II, 185-191 for historical 



remarks on the proofs by Lagrange and Euler]; Jour, de math. ele"m., (3), 5, 1891, 

 169-74; pamphlet, Paris, Nony, 1891. Reproduced by E. Humbert, Arithme'tique, 

 Paris, 1893, 284, and by G. Wertheim, Zeit. Math. Naturw. Unterricht, 22, 1891, 422-3. 

 86 Bull, des Sc. Math., (2), 15, I, 1891, 51-2. 



86 Math. Quest. Educ. Times, 57, 1892, 120-2. 



87 Jour, de math, elem., (4), 2, 1893, 73-6. 



88 Jour, fur Math., 112, 1893, 29. 



89 M<m. Acad. Roy. Sc. Belgique, 52, 1893-4, 22-28. 



