230 HlSTOKY OF THE THEORY OF NUMBERS. [CHAP. VI 



He stated he had a rigorous proof. He stated Feb. 16, 1745 (p. 312) that 

 it has not yet been proved that the sum of the squares of two relatively 

 prime integers has no divisor other than a El, nor that every prune 4n + 1 

 is a El, uniquely. 



Chr. Goldbach 23 proved Fermat's statement that a prime 4k I cannot 

 divide the sum of two relatively prime squares. Let a 2 be the minimum 

 square of the form (4n l)ra 1. Set v = 4n 1. Then 



v(m 2a + v) 1 = (a p) 2 , 

 so that a 2 ^ (a i>) 2 > whence v ^ 2a. Similarly, 

 {4(/i -a-fra)-l}m-l = (a- 2m) 2 , a 2 ^ (a - 2m) 2 , m ^ a. 



Thus a 2 + 1 = ?m ^ 2am ^ 2a 2 , a = or 1, values leading to contradic- 

 tions. 



Euler 24 prpved the Lemma: Every divisor of the sum of two relatively 

 prime squares is itself the sum of two squares. 



It is first shown that, if p = c 2 + d 2 is a prime and pq = a? + & 2 , then 

 q is a El. Since c 2 (a 2 + 6 2 ) - a 2 (c 2 + d 2 ) is divisible by p, one of the factors 

 be db ad is of the form rap. Set b = rac + x, a = db md + y. Then 

 ex d= dy = 0. But c is prime to d. Thus* x = nd, y = =F nc. Hence 



pg = (ra 2 + n 2 )(c 2 + d 2 ), q = ra 2 + n 2 . 



It now follows from (1) that, if the primes p if - , Pk and the product 

 Pi " p k q are all El, then q is a El. Hence if pq, but not q, is a El, p has a 

 prime factor not a El . 



Let p divide a 2 + 6 2 , where a, 6 are relatively prime, while p is not 

 a El. Set a = rap c, 6 = np db d, ^ c ^ p, ^ d ^ %p. Then 

 C 2 _f- d 2 _ pg < i^ Hence q has a prime factor r ^ p, not a (2. As 

 before, the divisor r of c 2 + d 2 divides a sum e 2 + ^ ^ |r 2 , and e 2 + / 2 

 has a prime factor ^ ^r not a El. Proceeding in this manner we ulti- 

 mately reach a contradiction with the fact that the sum of two sufficiently 

 small squares has all its prime factors sums of two squares. 



Euler gave a " tentative proof " of Girard's theorem that every prime 

 p = 4 n -}- i is a E] . If neither a nor 6 is divisible by p, a 4 " n is divisible 

 by p. If p divides the factor a 2 " + 6 2n , a El, then p is a El. It remains 

 to showf that a 2n 6 2n is not divisible by p for some pair of values of a, 6 

 [proved later by Euler 25 ]. 



Since p = a 2 + b 2 implies 2p = (a + 6) 2 + (a - 6) 2 , and conversely 

 2p = a 2 + 6 2 implies p = a 2 + /3 2 , where a = (a + 6)/2, /3 = (a - 6)/2 are 

 integers, there are as many representations of p as of 2p as a sum of two 

 squares (including the case in which one square is zero). 



23 Correep. M*th. Phys. (ed., Fuss), 1, 1843, 255, letter to Euler, Sept. 28, 1743. Euler, p. 



258, expressed surprise at the simplicity of the proof. 

 M Ibid., 416-9; letter to Goldbach,' May 6, 1747. Novi Comm. Acad. Petrop., 4, 1752-3 



(1749), 3-40; Comm. Arith., I, 155-173. 

 * In the letter, it is concluded from be db ad = m (c 2 + d z ) that md T a is divisible by c; 



Thus T a = en dm, b = cm + dn. 

 t In the letter, it is stated that there are innumerable cases in which a 2n 6 2 " is not divisible 



by 4?i + 1. 



