CHAP. Xlli] SOLUTION OF x' 1 ~y' 1 = g. 403 



if A is not a multiple of 4. For A = 4B, the sum equals A if h = Bfff; 

 then x 2 +A = (/i+2/) 2 for x = h. Next, let A = 2B+l. The sum of 2/+1 

 consecutive differences 2/1+1, is (2/+l)(2/&+2/+l), which can be 

 made equal to A by choice of h, whence x 1 A = h? if 



B+f+l 



T. Clowes 18 noted that the difference of the squares of x+l and x l 

 equals the difference of the squares of a+6 and a b if x = ab. 



L. Poinsot 19 stated that any integer N, not the double of an odd integer, 

 can be represented as a difference of two squares and in as many ways n 

 as N can be expressed as a product of two factors both odd and relatively 

 prime or both even and with no common factor > 2. If N has k distinct 

 prime factors, n = 2 k ~ 1 . 



P. Volpicelli 20 took g = 2 M /it hi, where the /i's are distinct primes. 

 As known, the number of decompositions of g into two factors is 



or v-\-% according as at least one of the exponents p, a, , T is odd or all 

 are even. Hence, in the respective cases, the number of decompositions 

 into two distinct even factors, i. e., the number of solutions of x* y* = g, is 



or PI %, if ju>0. For ju = 0, the number of solutions is v or v \, respec- 

 tively. 



R. P. L. Claude- 1 noted that any odd integer ={=1 is a difference of 

 two squares since ab is the difference of the squares of (a6)/2, while the 

 double of an odd integer is not. Every integer which is a difference of two 

 squares is such as many times as there are different combinations 2,3, , n 

 at a time of its n prime factors. 



G. C. Gerono 22 stated only known results. 



L. Lorenz 23 concluded from 



TO, n= oo m=l n=l 



that the number of solutions of m 2 n? = N is double the number of divi- 

 sors of N or A T /4 according as N is odd or is divisible by 4; none if N/2 

 is odd. 



G. H. Hopkins 24 noted that in x z y z =(2a l - -a n ) 2 , where a i} , a n 

 are primes, x or y has (3 71 1)/2 integral values. 



18 The Ladies' and Gentlemen's Diary (ed., M. Nash), New York, 3, 1822, 53-4. 



19 Comptes Rendus Paris, 28, 1849, 582. 



20 Atti Accad. Pont. Nuovi Lincei, 6, 1852-3, 91-103; Annali di So. Mat. e Fis., 6, 1855, 



120-8; Comptes Rendus Paris, 40, 1855, 1150; Nouv. Ann. Math., 14, 1855, 314. 



21 Nouv. Ann. Math., (2), 2, 1863, 88-90. 



22 Ibid., 90-92. 



23 Tidsskrift for Math., (3), 1, 1871, 113-4. 

 24 Math. Quest. Educ. Times, 16, 1872, 46-7. 



