On the number of Solutions of a Diophantine equation of Frobenius

Abstract

We consider a linear Diophantine equation of the form x\a\ + ...+Jt"0" -N t where n is a fixed integer greater than one, 0 < a\ < ... < a n is a fixed set of integers such that (a\, ...,a n ) = 1. We denote by/(W) the number of solutions in non-negative integers. It is well known that/Cx) = P(x) + A(JC), where P(x) is a polynomial in χ of degree η -1 and Δ(*) is a periodic function with period a\ ...a". We apply an elementary approach to the problem of calculating Δ(*), and utilize roots of unity arguments in constructing this periodic function. For/(N), an explicit expression is obtained for arbitrary Λ; this expression includes complicated sums containing the roots of unity. In the case π = 2, this approach leads to a computable explicit expression for/(jc). We note that previously the expression for Δ(*) has not been known.

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