476 HISTORY OF THE THEORY OF NUMBERS. [CHAP, xvi 



if (F 1 ) mg*-nV = Nk*, m- l Ah*-rr l Bg* = Nf*, or if (F 2 ) 

 nf 2 -}-nr l Ak 2 = Ng z . Hence the solution of (F) can be derived from the 

 solution of (Fi) or (F 2 ). Giving suitabte values to m, n, N, A, B, we can 

 readily derive all of Collins' formulas from (Fi) and (F 2 ). 



A. Genocchi 86 stated that x 2 -\-h and x 2 -\-k are not both squares if (i) 

 h 1, k a prime or square of a prime 8m=b3, provided the odd prime factors 

 of k 1 are all of the form 4n+3; (ii) h = 2, k a, prime 8m-\-3 or double of a 

 prime 8m+5, provided the odd prime factors of k 2 are all of the form 

 8n +7; (iii) h a prime 8m 3, k a prime 8m +7, provided the odd prime 

 factors of h k are all of the form 4ft +3 and quadratic non-residues of k; 

 (iv) h a prime 8m +3, k = h 2 , provided the odd prime factors of h 1 are all 

 of the form 4ft +3 ; (v) h a prime, k = hp, where h and p are primes one of the 

 form 8m +3 and the other of the form 8m +7, provided the prime factors 

 of p 1 other than 2 and h are all of the form 4ft +3 and quadratic non-resi- 

 dues of h. 



A. Genocchi 87 treated (1) by the method of Diophantus: Set r = mx+p, 

 s = nx p. Then bzq z = (r 2 p z }b(a, so that the second equation (1) becomes 



2p(an+bm) 



x= 



p being given in the problem of Diophantus II, 17. In Euler's 76 problem, p 

 is unknown; the first of the preceding equations determines p in terms of 

 ra, n, X] then azq~ = amnx-(m-\-ri)/ '(an + 6m). By setting n = bl, x = 2(m-\-al), 

 we get formulas derived from Euler's by changing p, I, m into p, y, v. 

 Genocchi noted that the present problem is equivalent to that of solving 

 y' 2 x 2 :z 2 y 2 = a:b, treated fully by Leonardo Pisano. 6 For, (1) gives 

 r 2 p 2 : s 2 p z = a : 6, and conversely, if we set r 2 p 2 azq 2 . Genocchi 

 proved (pp. 9-23) that the system x 2 +a=D, 2 +6=D is impossible in 

 rational numbers for a= 1 and b a prime 8fc3 such that 6 1 has no prime 

 divisor 4Z+1 (as 6 = 3, 5, 13, 19, 29, 37, 43); for a = 2 and b a prime 8/v+3 

 such that every divisor of 62 is of the form 8^+7 (as 6 = 163, 331, 449); 

 a = 2, 6 = 2A, A a prime S&+5 such that A I has no odd prime divisor not 

 of the form 8t+7 (as A = 5, 29, 197, 317); a = A, 6 = A5, where A, B are 

 primes, one of the form Sk-\-7 and the other 8A;+3, such that A is a quadratic 

 residue of B when A=8k-\-7, B l has no odd prime divisor 4-j-l not a 

 quadratic residue of A, and, in case A = 8/j+3, (B l}/2 is divisible by A if 

 a quadratic residue of A (as A = 3, 5 = 7; A = 7, 5 = 3 or 19; A = 11, 5 = 23); 

 a a prime 8fc+3, every odd prime divisor of a I being of the form 4+3, 

 6 = a 2 ; a=l, 6= 8 or the negative of a prime 8fc3 or the square of a 

 prime 8fc3, no prime divisor of 6 1 being of the form 4Z+1 in the third 

 case. 



The following three papers relate to the system t~+u? = 2v~, P-{-2u 2 = 3w z . 



E. Lucas 88 treated the equivalent system 2y 2 v? = t 2 , 2v 2 +w 2 = 3w; 2 and 

 showed how to get new solutions from one. [Cf. Pepin. 90 ] 



86 Comptes Rendus Paris, 78, 1874, 433-5. 



87 Memorie di Mat. e Fis. Soc. Italiana Sc., (3), 4, 1882, No. 3. 



88 Nouv. Ann. Math., (2), 16, 1877, 409-416. 



