242 HISTORY OF THE THEORY OF NUMBERS. [CHAP. VI 



primes 4k + 1. To find a, b without trial, express p, p' as HI by continued 

 fractions and apply (1) and 



(P2 + Q2)< = 2 + fJ2 } 



H = tp^Q - 



G. L. Dirichlet 88 used the theory of binary quadratic forms to prove 

 that, if m is a product of powers of n primes 4h + 1, the number of sets of 

 relatively prime solutions x, y of x 2 + y* = m is 2' x+2 . The number ( 91) 

 of all sets of solutions is the quadruple of the excess of the number of its 

 divisors 4h + 1 over the number of its divisors 4h + 3. 



A. Vermehren, 89 to express z 3 as a sum of two squares, put z = u -f v; 

 then z 3 = u?(u + 3v) + v 2 (3it + v). He took u + 3y = 4n 2 , 3u + v = 4m 2 . 



F. Unferdinger 90 noted that the product of the expansions of (a db bi) m 

 gives (a 2 + 6 2 ) m = A 2 + J5 2 , where A, 5 are known polynomials. He 84 

 had shown that a product P of n sums of two squares can be expressed as a 

 E) in 2 71 " 1 ways distinct in general. The same result therefore holds for P m . 



G. C. Gerono 90a proved that every divisor of a sum of two relatively 

 prime squares is a sum of two relatively prime squares. 



V. Eugenio 91 proved the Lemma 24 as follows. Let M divide P 2 + Q 2 , 

 where P is prime to Q, and call P'/Q' the next to the last convergent of the 

 continued fraction for P/Q. Then PQ' - P'Q = d= 1. By (1), M divides 

 (PP f + QQ'Y + 1. Thus M divides AT 2 + 1, where N is an integer < M. 

 Express M/N as a continued fraction with an even number of quotients: 



where n = 2s. Let Mi/Ni, , M n /N n = M/N be the successive con- 

 vergents. Then 



(4) M i+l 



-i a n -2 + + ai + a 



Now N 2 + 1 = MN'. Thus by (4 8 ), -MW' - JV n _i) = N(N - Mn-i). 

 Thus M divides A/" - M n _i < M. Hence Jlf n _i = N. Thus (5) equals 

 f, and a = a n -i, etc. Hence 



= l J 



: " 



__ _ 



" ~ 



But M_! = N 8 . Thus M/AT = (Jf! + N 2 a )/M s N s , M = M] + Nl 



88 Zahlentheorie, 68, 1863; ed. 2, 1871; ed. 3, 1879; ed. 4, 1894. 



89 Die Pythagoraischen Zahlen, Progr. Domschule, Gtistrow, 1863. 



90 Archiv Math. Phys., 49, 1869, 116-7. 



900 Nouv. Ann. Math., (2), 8, 1869, 454-6, 559. 



91 Giornale di Mat., 8, 1870, 162-5. 



