CHAP. II] 



NUMBER OF SOLUTIONS OF ax+by = 



69 



V. Bernard! 135 would find the positive integral solutions of ax + by = k 

 by employing the remainders r[, r[' and quotients q[, q\' obtained on dividing 

 k b by a and k a by b. Thus 



axi + byi = ki t ki = k a b r( r". 



Similarly, ax 2 + by 2 = fca, , ax m + by m = km = k m -i a b r' m r^, 

 where r' m , r'^ are the remainders and q m , q^. the quotients obtained on 

 dividing k m -i b by a and k m -i a by b. In this way we find a value 

 uoim such that a zero remainder results from that one of the two divisions 

 in which the divisor is the smaller of a, b, or such that the remainder from 

 the other division is zero or is divisible by the smaller coefficient. Then 

 k u is divisible by the larger or the smaller of a, b in the respective cases. 

 The positive integral solutions with k u divisible by a are 



x u = k u /a nb, y u = na (n = 0, 1, 

 Then all positive integral solutions of the given equation are 



= ( 



x u + q[ - 



Cf. Hermite. 119 



L. Crocchi 136 noted that Hermite's 119 formulas do not give merely the 

 integral solutions. Thus, if n < a, n < b, they give x = b, y = =fc ar], 

 % + 77 = n/(ab), which lead to fractional solutions of ax + by = n. 

 Crocchi therefore transformed Hermite's formulas so that the resulting 

 formulas give merely positive integral solutions. Set 



n' = n r s, 



Then 



1= ? .-= ^.. - =^ 



a" LaJ a La \ La J a' La J LaJ 



where [s/a] + is the quotient by excess of s by a. Similarly, 



r^i = [?]_[!] _[] . 



L a J La J LaJ+ La J+ 



Taking alternate signs and adding, we get, for m even, . 



LaJ 1LJ + "LTJ + 



a+ 



+ 



135 Atti society italiana per il progresso delle scienze, 2, 1908, 317-8. 

 II Boll, di Matematica Gior. Se.-Didat., 7, 1908, 229-236. 



