Recurrence relations for the number of solutions of a class of Diophantine equations

TL;DR

This paper develops recursive formulas for counting solutions to various Diophantine equations, extending to non-linear cases, and expresses solutions explicitly using Bell polynomials based on generating functions.

Contribution

It introduces a unified recursive approach for linear, quadratic, and non-linear Diophantine equations with explicit solutions via Bell polynomials.

Findings

Recursive formulas for solution counts derived

Explicit solutions expressed through Bell polynomials

Applicable to linear, quadratic, and non-linear equations

Abstract

Recursive formulas are derived for the number of solutions of linear and quadratic Diophantine equations with positive coefficients. This result is further extended to general non-linear additive Diophantine equations. It is shown that all three types of the recursion admit an explicit solution in the form of complete Bell polynomial, depending on the coefficients of the power series expansion of the logarithm of the generating functions for the sequences of individual terms in the Diophantine equations.