66 HISTORY OF THE THEORY OF NUMBERS. [CHAP, n 



J. J. Sylvester 122 stated two theorems on the number (n; a, b) of positive 

 integral solutions of ax + by = r for the values r = 0, 1, , n: 



(n; a, 6) = %k(kdb + a + b + 2n' - 1) + (n'; a, b), 

 if k and n' are positive integers for which n + 1 = kab + n r ' } 



, M ,:>w\ 



(-;, 6) = ("';', z-') 



where a', &' are positive integers such that ab f ba' = 1, a' < a, b' < b. 

 E. Catalan 123 made use of the known fact that the solutions of 



ax + by = n 



are x = a b6, y = /3 + ad, if a, /3 is one set of positive integral solutions. 

 Let a, 6, n be positive. Then the positive solutions have B < <x/b, 6 > ft fa, 

 which are equivalent to 6 < aa/(ab), 6 > (aa ri)/(ab). Hence co = [nlaH] 

 or [n/ab~]+l. Writing n = abq+n', Q^ri<ab, he proved that ax+by = n 

 has <? + 1 or q positive solutions according as ax' + by' = n' has a positive 

 solution or none. 



C. de Polignac 124 remarked that ax + by n may be solved graphically 

 by means of a lattice whose initial rectangle has the base a and altitude b. 

 He concluded that, if T = [n/ab'], u> = T if the remainder obtained by 

 dividing n by ab is < b/3, where /3 is the least positive y, while co = r + 1 

 in the contrary case. 



E. Catalan 125 stated and E. Cesaro proved that, if we count the integral 

 solutions ^ of each of the equations x + 2y = n 1, 2x + 3y = n 3, 

 3x + 4?/ = n 5, , the total number of solutions equals the excess of 

 n+2 over the number of divisors of n+2. For, px+(p-\-l)y = n (2p 1) 

 has 



r + n 



L J 



solutions == 0, where e = 1 or according as p + 1 is or is not a divisor 

 of n + 2. 



E. Cesaro stated and J. Gillet 126 proved that if we count the integral 

 solutions ^ of each of the equations x + 4y = 3n 1, 4x + 9y = 5n 4, 

 9x + 16?/ = 7n 9, , the total number of solutions is n. 



E. Catalan stated and E. Cesaro and H. Schoentjes 127 proved that if 

 we count the integral solutions ^ of each of the n + 1 equations 



122 Comptes Rendus Paris, 50, 1860, 367; Coll. Math. Papers, II, 176. 



128 Melanges Math., 1868, 21-23; Mem. Soc. Sc. Liege, (2), 12, 1885, 23 (Melanges Math. I). 



Mathesis, 10, 1890, 220-2. 

 124 Bull. Math. Soc. France, 6, 1877-8, 158. E. M. Laquiere, ibid., 7, 1878-9, 89, simplified 



Polignac's work. A resume 1 of both is given by S. Giinther, Zeitschr. Math. Naturw. 



Unterricht, 13, 1882, 98-101. 



B Nouv. Ann. Math., (3), 1, 1882, 528; (3), 2, 1883, 380-2. 

 128 Mathesis, 2, 1882, 208; 5, 1885, 59-60. 

 lUd., 2, 1882, 158; 3, 1883, 87-91. 



