CHAP. II] NUMBER OF SOLUTIONS OF ax+by = n. 65 



Let henceforth a and b be relatively prime and positive. Let /3 denote 

 the least positive integer such that n a(3 is divisible by 6. Then every 

 solution is given by 



, n a/3 



x = /3 + bm, y = = am. 



b 



The values of m making x and y positive are 0, 1, -, E, where E is the 

 largest integer less than (n a/3)/(a&). Thus there are co = E + 1 sets 

 of positive integral solutions x, y. 



P. Barlow 118 employed positive integers p, q such that aq bp = + 1. 

 Then all solutions of ax + by = n are given by 



x = nq mb, y = ma np. 

 Let [f\ denote the greatest integer ^ t. Then 



a 



or one less according as nq/b is not or is an integer. In fact, m must be 

 less than nq/b and > np/a to make x and y positive. 



Libri 27 expressed co as a sum of trigonometric functions. 



C. Hermite 119 employed the integers 



Then every positive integral solution of ax + by = n is given by 





ft] -KM?]-- 



where , 77 take the co + 1 sets of integral values ^ which satisfy + y = . 

 Here co is such that n (i) uab. Thus if T is the greatest integer =i n/(ab), 

 and n = ro& + u, then co = r or r + 1, according as az + by = v has 

 positive integral solutions or not. 



M. A. Stern 120 gave Barlow's 118 result. 



A. D. Wheeler 121 noted that if ax-\-by = c has the least positive solution 

 x = v, it has the solutions x = v + b, etc., and hence n positive solutions 

 if c > nab. The least and greatest values of c for n positive solutions are 

 (n l}ab + a + b and (n + l)ab. If c = n6 there are exactly n 1 

 solutions. If c = nab + ax' + &?/, there are n + 1 solutions. 



118 Theory of Numbers, London, 1811, 324. 



119 Quar. Jour. Math., 1, 1855-7, 370-3; Nouv. Ann. Math., 17, 1858, 127-130. Oeuvres, 

 I, 440. Cf. Crocchi. 136 



120 Jour, fur Math., 55, 1858, 210. 



121 The Math. Monthly (ed., Runkle), New York, 2, 1860, 56, 193-4. 



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