CHAP. II] NUMBER OF SOLUTIONS OF ax+by = n. 67 



x + 2y = n, 2x + 3y = n - 1, , (n + l)x + (n + 2)y = 0, the total 

 number of solutions is n + 1. 



Cesaro 128 proved the last theorem with n replaced by n 1, by showing 

 that p(x + y + l)+y = n has exactly N p = [n/p] [nf(p + 1)] in- 

 tegral solutions ^ 0. Also, 



while NI + Nz 4- N 5 + equals the difference between the number of 

 odd divisors and the number of even divisors of 1, 2, , n. The number 

 of integral solutions ^ of x + 2y = 2(n 1), 2x + 3y = 2(n 2), , 

 nx + (n + l)y = is the number of non-divisors of 2n + 1. As a general- 

 ization, px + (p + I)?/ = fc( w p), for p = 1, , n, have 



J|f = J|f 1 + + M n 



integral solutions ^ 0, where M p [_kn]p~] [(kn + k l)/(p + 1)] is the 

 number of solutions of the equation written; for k = 3, M equals the sum 

 of the numbers of divisors of Bn + 1 and 3n + 2. [The preceding results 

 are special cases of a formula given by Lerch in 1888; cf. Gegenbauer, 29 

 p. 227 of Vol. I of this History.] As a generalization of Catalan's 127 

 theorem, the total number of integral solutions ^ of 



(1 + jk)x + (l+j + lk)y = k(n - j - 1) (j = 0, 1, ) 



is n. Given a set x = a, y = ft of integral solutions of ax + by = n, 

 the number of integral solutions ^ is [fila~] [_(a !)/&]. 



Consider a set HI, Vv, - - - of positive integers each prime to the term 

 following it. Let v i} v 2 , be integers and determine a series of w's by 



w p = v p u p+ i 



If w r is the first negative term, the total number of integral solutions ^ of 



u p x + u p+l y = w p (p = 1, , r - 1) 



is [VI/MI] [y r /w r ], since the equation written has [vpju p ~\ 

 solutions ^ 0. The case v p = n, u p = p 2 , gives the result of Cesaro. 126 

 He quoted (p. 273) from a letter from Hermite the result that 



each member being the number /x of sets of positive integers for which 

 ax + by si n. Henceforth, let a and b be relatively prime. Then the 

 number of integral solutions ^0 of ax + by n is known to be 

 N n = [nldb~\ + r, where r = or 1. Cesaro noted (p. 278) that r = 1 if 

 the remainder R obtained by dividing n by ab is of the form pa + crb, 

 where p, a are integers i= 0, and r = in the contrary case [Catalan 123 ]. 

 This theorem, which may be expressed in the form A^ n N R = [n/ab~], is 



" 8 M6m. Soc. Roy. Sc. de Ltege, (2), 10, 1883, 263-283. 



