586 HISTORY OF THE THEORY OF NUMBERS. [CHAP, xxi 



where d = n 2 1+p 2 . The general solution thus involves the rational 

 parameters p, t. 



E. Catalan 270 stated that, if r = l, integral solutions of (1) are 



where k = l or 2, while a, b are relatively prime integers. For example, 

 if a = 5, 6 = 1, we may take 7 = 313, w = 7850 (in place of 1850 in Table X in 

 Legendre's Theorie des nombres), whence n = 626, = 3600. 



Catalan, 271 in treating the integral solutions of (1) for r = l, wrote 

 a = 2ns, 13 = ff, where s, a are given by Lebesgue's 249 (3) for r = l. The 

 problem is then to make a/3 the square IQy 2 of an integer. Since sn is 

 even, y will then be an integer. But his separation into two cases lacks 

 generality and his solution is incomplete. His 272 later discussion leads to the 

 following result : Take any two relatively prime integers p, q, one even, and 

 express pq/2 as a product of a square u' 2 by a number 6 without a square 

 factor; then if 



has integral solutions 7, v, we have 



M. Cantor 273 reported on Catalan's 271 - 2 discussion of the preceding equa- 

 tion a(3 = 16y 2 , where a and /? are integers divisible by 4 for which /3d=a+l 

 are squares, and obtained two sets of solutions, in which p and q are rela- 

 tively prime integers, one an odd square and the other either half of an even 

 square or an even square. In the first case, (p 2 -\-q z )y 2 u 2 = 1 yields integers 

 7, u, and then y 2 = 2pq(yu/4:Y. In the second case, if (p 2 -\-q 2 )y 2 2w 2 = l 

 has integral solutions 7, w, then y 2 pq(yw/2) 2 . In each case, n = py, 



C. Richaud 274 treated (1) for r = l, viz., l 2 k 2 = y 2 , where 2k = x(x 1), 

 2l=(x j t-ri)(x+n 1). Certain, but not all, solutions arise from Z = a 2 +6 2 ; 

 fc, y = 2ab, a 2 6 2 . Eliminating #, ?/, we get a quartic equation. For 

 k = 2ab, it becomes 



m 2 -(4 4 +l)n 2 =-l, m = 2(a+6), nt = a-b, 



with an infinitude of solutions ?w = 2 2 , n = 1 ; m = 32i 6 +6 2 , n = 16 4 + 1 ; etc. 

 Note that the sum of the numbers x, x+1, , x+n 1 is a square, (a 6) 2 . 

 For a general r, (1) becomes nsa = Sy 2 by Lebesgue's 249 (3). For 276 ns/2 = 6 2 , 

 <7/4 = aa 2 , y = aab, he eliminated s and discussed at length the resulting 



270 Bull. Acad. Roy. de Belgique, (2), 22, 1866, 339-40. 



271 Atti Accad. Pont. Nuovi Lincei, 20, 1866-7, 1-i; Nouv. Ann. Math., (2), 6, 1867, 63-67; 



Melanges Math., 1868, 99-103. 



272 Atti Accad. Pont. Nuovi Lincei, 20, 1866-7, 77; Nouv. Ann. Math., (2), 6, 1867, 276-8; 



Melanges Math., 1868, 248-251. 



273 Zcitschr. Math. Phys., 12, 1867, 170-2. 



274 Atti Accad. Pont. Nuovi Lincei, 20, 1866-7, 91-110. 



276 In the alternative case ns/4 = e*6 2 , <r/2 = aa 2 , y = aab, not treated, there are two misprints 

 for 4. 



