Journal of the Ramanujan Mathematical Society

Volume 41, Issue 3, September 2026 pp. 233–251.

Frobenius numbers associated with diophantine triples of x{2} + y{2} = z{r} for r = 4,5,6

Authors: Takao Komatsu, Neha Gupta and Manoj Upreti
Author institution:Department of Mathematics, Institute of Science Tokyo, Tokyo 152-8551, Japan.

Summary: For given positive integers a{1}, a{2}, …, a{k} such that gcd (a{1}, a{2}, …, a{k}) = 1, the Frobenius number is the largest integer that cannot be written as nonnegative integer combination of a{1}, a{2}, …, a{k}. In this article, we will calculate the Frobenius number for triples (a,b,c) where a,b, and c are coprime positive integers that satisfy the Diophantine equation x{2} + y{2} = z{r} for r = 4, 5, 6. This work extends our previous work, which focused on the case r = 3. Our method is characterized by finding the Frobenius number by geometrically arranging all elements of the Apéry set and determining the largest element. We impose certain additional hypotheses to obtain our results. Finally, we extend the discussion to outline a general framework applicable to arbitrary values of r.

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