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arXiv · 2106.13796

Some Bounds for Number of Solutions to $ax + by + cz = n$ and their Applications

Damanvir Singh Binner
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Abstract

In a recent work, the present author developed an efficient method to find the number of solutions of $ax+by+cz=n$ in non-negative integer triples $(x,y,z)$ where $a,b,c$ and $n$ are given natural numbers. In this note, we use that formula to obtain some simple looking bounds for the number of solutions of $ax+by+cz=n$. Using these bounds, we solve some special cases of a problem related to the generalization of Frobenius coin problem in three variables. Moreover, we use these bounds to disprove a recent conjecture of He, Shiue and Venkat regarding the solution structure of $ax+by+cz=n$.

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Damanvir Singh Binner. 2021-06-25. Some Bounds for Number of Solutions to $ax + by + cz = n$ and their Applications. https://arxiv.org/abs/2106.13796

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