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Frobenius numbers associated with Diophantine triples of $x^2+y^2=z^r$ (extended version)

Takao Komatsu, Neha Gupta, Manoj Upreti

Abstract

We give an explicit formula for the $p$-Frobenius number of triples associated with Diophantine equations $x^2+y^2=z^r$, that is, the largest positive integer that can only be represented in $p$ ways by combining the three integers of the solutions of Diophantine equations $x^2+y^2=z^r$. When $r=2$, the Frobenius number has already been given.

Frobenius numbers associated with Diophantine triples of $x^2+y^2=z^r$ (extended version)

Abstract

We give an explicit formula for the -Frobenius number of triples associated with Diophantine equations , that is, the largest positive integer that can only be represented in ways by combining the three integers of the solutions of Diophantine equations . When , the Frobenius number has already been given.
Paper Structure (28 sections, 7 theorems, 134 equations, 16 tables)

This paper contains 28 sections, 7 theorems, 134 equations, 16 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. $x^2+y^2=z^3$
  4. The case where $\sqrt{3}t<s<(2+\sqrt{3})t$
  5. The case where $\mathbf x<\mathbf z<\mathbf y$
  6. The case $s>(2+\sqrt{3})t$
  7. The case for $r=4$
  8. The case where $s>\left(1+\sqrt{2}+\sqrt{2(2+\sqrt{2})}\right)t$
  9. The case where $(1+\sqrt{2})t<s<\left(1+\sqrt{2}+\sqrt{2(2+\sqrt{2})}\right)t$
  10. The case for $r=5$
  11. The case where $s>\left(1+\sqrt{5}+\sqrt{5+2\sqrt{5}}\right)t$
  12. The case where $\sqrt{5+2\sqrt{5}}\,t<s<\left(1+\sqrt{5}+\sqrt{5+2\sqrt{5}}\right)t$
  13. The case where $\left(1+\sqrt{5}-\sqrt{5+2\sqrt{5}}\right)t<s<\sqrt{1-2/\sqrt{5}}\, t$
  14. The case where $0<s<\left(1+\sqrt{5}-\sqrt{5+2\sqrt{5}}\right)t$
  15. Sylvester number (genus)
  16. ...and 13 more sections

Key Result

Lemma 1

Let $k$ and $p$ be integers with $k\ge 2$ and $p\ge 0$. Assume that $\gcd(a_1,a_2,\dots,a_k)=1$. We have

Theorems & Definitions (8)

  • Definition 1
  • Lemma 1
  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Theorem 5
  • Theorem 6