Analogues of the Frobenius problem (Q1761039)

The article under review is devoted to the equation \[ ax+by=M,\tag{1} \] where all the parameters \(a\), \(b\), and \(M\) and the variables \(x\) and \(y\) are positive integers (the Diophantine Frobenius problem). The author consider equation (1) under the additional constraints \[ (x,m)=(y,n)=1,\tag{2} \] where \(m\) and \(n\) are fixed positive integers. In the first theorem it is proved that if \((am,bn)=1\) then equation (1) with restrictions (2) is solvable for \(M>M_0(a,b,m,n)\). Theorem 2 contains a formula for the number of solutions of the problem (1)--(2). This number is expressed in terms of Dedekind sums. Theorem 3 gives an upper bound for \(M_0(a,b,m,n)\).