CHAP. VI] SUM OF TWO SQUARES. 231 



From Girard's theorem and (1) it was concluded that any number is 

 a E] if it has the form 2 y a 2 6, where each prime factor of b is of the form 



4k + 1. 



Euler 25 later succeeded in establishing the point which he could not 

 prove in his preceding paper. 24 If the differences (a + l) 2n a 2 " of the 

 first order of 1, 2 2n , 3 2n , -, (4n) 2n were all divisible by p, the differences of 

 order 2n would be divisible by p, whereas they equal (2n)!. This point 

 can also be proved by means of Euler's 26 criterion for quadratic residues; 

 however, Euler proved this criterion by the method of differences. In the 

 former 25 paper ( 70), Euler noted that the negative of a residue of a square 

 when divided by a prime 4n 1 is not the residue of a square, whence 

 o 2 + 6 2 is not divisible by 4n 1 if a and 6 are not. Since a product of 

 primes of the form 4Jc + 1 is of that form, it follows ( 73) that 4n 1, 

 whether prime or composite, is not a divisor of a sum of two relatively prime 

 squares. 



Lagrange 9 of Ch. VIII proved that if a E] divides a E] the quotient is a El. 



Euler 27 proved (1) by multiplying (a + 6i)(c + di) by its conjugate. 



Euler 28 gave a more elegant proof of the Lemma. 24 Let N divide 

 P 2 + Q 2 , where P and Q are relatively prune. Set 



P = fN p, Q = gNq, O^p^^N, O^q^^N. 



Then p 2 + q 2 = Nn, where n ^ %N. Set p = an + a, q = fin + b, where 

 a and b are numerically ^ \n. Set A = aa + 6/3. Then 



Nn = n z (a* + /3 2 ) + 2nA + a? + b 2 . 



Hence a 2 + 6 2 = nn', n' ^ Jn. Thus N = n(a 2 + /3 2 ) + 2 A + n'. By (1), 

 nn'(c? + /3 2 ) = (a 2 + 6 2 )(a 2 + /3 2 ) = A 2 + 2 , = aft - ba. 



Hence JW = (n r -f ^4.) 2 + B 2 . Just as this was derived from Nn = p 2 + q* f 

 so from it we get Nn" = E], n" ^ \n' , etc., finally N-l - EL 



C. G. J. Jacobi 1 (p. 341) repeated this proof and stated that, while it 

 contained nothing not known to Diophantus, there is no ground for the 

 assumption that the latter actually possessed the proof. 



Euler 29 gave a second proof of Girard's theorem. Since 1 is a 

 quadratic residue of every prime p = 4n + 1, there exists a square b z 

 with the residue 1, so that p divides 1 + 6 2 . Hence, by the Lemma, 

 p is a E] . 



In a posthumous manuscript, Euler 30 proved the first step in the above 

 Lemma. Let P = p 2 + q z be divisible by A = a? + fe 2 , where a is prime 



25 Corresp. Math. Phys. (ed., Fuss), 1, 1843, 493; letter to Goldbach, April 12, 1749. Novi 



Comm. Acad. Petrop., 5, 1754-5 (1751), 3; Comm. Arith., I, 210. 



26 Novi Comm. Acad. Petrop., 7, 1758-9 (1755), 49, seq., 78; Comm. Arith., I, 273. 



27 Algebra, St. Petersburg, 2, 1770, 168-172. French transl., Lyon, 2, 1774, pp. 201-8. 



Opera Omnia, (1), I, 417-420. 

 28 Acta Eruditorum Lips., 1773, 193; Acta Acad. Petrop., I, 2, 1780 (1772), 48; Comm. 



Arith., I, 540. Proof reproduced by Weber-Wellstein, Encyklopadie der Elem. Math., I 



(Alg. und Analysis), 1903, 244-250. 



29 Opusc. anal., 1, 1783 (1772), p. 64 seq., 36; Comm. Arith., I, 483. 

 80 Tractatus de numerorum, 564-7; Comm. Arith., II, 572. Same in Opera Postuma, 1, 



1862, 72. 



