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



S. Roberts 55 proved the known result that if x 2 Py 2 are squares then 

 P?/ 2 is of the form tab(a- b 2 ), where = 4 or 1 according as a, b are of 

 different or like parity. He stated that the values of P which are primes 

 or the doubles of primes are all obtained by the rule of Leonardo which 

 makes three of a, b, ab squares, and carried further the analysis of 

 Genocchi. 43 Inadmissible values of P are primes Sfc+3 or doubles of 

 primes 8fc+5. He proved that various classes of primes P are excluded, 

 all being such that x 2 2Py 2 = 1 has no solution. 



A. Desboves 56 started with a congruent number Xju(X 2 ju 2 ), changed X 

 to x 2 , fji to ?/, absorbed the factor x-y~ into the term Y 2 of X 2 ^aY 2 = D, 

 and obtained the congruent number x 4 7/ 4 . Since a 4 +d 4 = 6 4 +c 4 is solvable 

 and hence also 



(a 4 - 6 4 ) (< - 6 4 ) = (ad) 4 - (be) 4 , 



we can find an infinitude of numbers which are differences of two bi- 

 quadrates and whose product is such a difference, and hence an infinitude 

 of solutions of Boncompagni's 48 problem to find two congruent numbers 

 whose product is a congruent number. 



A. Genocchi 57 proved that the following numbers are not congruent: 

 a prime 8k +3 or the product of two such primes; the double of a prime 

 8/c+5 or the double of the product of two such primes. 



Genocchi 58 stated his 44 results and that no congruent number is a 

 product of a square by a prime 8m+3, or by double a prime 8m+5, or by 

 a product of two primes 8ra+3, or by double the product of two primes 



G. Heppel 59 found a such that 101 2 +a = (101-h&) 2 , 101 2 -a = (101-Z) 2 

 by taking l = k+c, whence 2k 2 = 202c 2kc c 2 . Since c is a factor of 2k 2 , 

 but not of k, c = 4. Thus k = 18, a = 3960. 



M. Jenkins 60 found an integer a for which (m 2 +n 2 ) 2 a = (ht) 2 . Then 

 a = 2ht, (m 2 -\-n 2 ) 2 = h 2 -\-t 2 . One solution of the latter is h = m 2 n 2 , t = 2mn. 



G. B. Mathews 61 discussed x 2 d=a= D. From x 2 -\-a = (x+m) 2 , we get x. 

 Then x 2 -a = N/(4m 2 ), where JV = a 2 -6am 2 +?w 4 . Set N=(a-m 2 k/l) 2 . 

 Then m 2 =/a, f=2l(k-3l)f(k 2 -l 2 ). Take a=fb 2 , where b is arbitrary. 

 Then m=fb and x is found. 



R. Aiyar 62 noted that, if A 2 4B are squares, A and B are expressible 

 in one and but one w r ay in the forms A=X(ra 2 -|-n 2 ), B = \ 2 mn(m 2 n 2 ), 

 where m and n are relatively prime and one is even. 



A. Cunningham and R. W. D. Christie 63 solved x 2 y 2 = y 2 w 2 cz 2 , 

 where c is given, as c = 65, by use of x 2 2y 2 = w 2 .- [Hence y 2 cz 2 = w 2 , 



M Proc. Lond. Math. Soc., 11, 1879-80, 35-14. 



66 Assoc. frang., 9, 1880, 242. 



67 Memorie di Mat. e Fis. Soc. Ital. Sc., (3), 4, 1882, No. 3. 



68 Nouv. Ann. Math., (3), 2, 1883, 309-10. 



69 Math. Quest. Educ. Times, 40, 1884, 119. 

 10 Ibid., 41, 1884,65-6. 



61 Ibid., 107-8. 



62 Math. Quest. Educ. Times, 65, 1896, 100. 



63 Math. Quest. Educ. Times, (2), 13, 1908, 77-9. 



