On the divisibility of class numbers of quadratic fields and the solvability of Diophantine equations

TL;DR

This paper establishes criteria linking the divisibility of class numbers of quadratic fields to the solvability of specific Diophantine equations, advancing understanding of number theory and algebraic structures.

Contribution

It introduces new criteria for class number divisibility and Diophantine equation solvability, connecting algebraic number theory with Diophantine analysis.

Findings

Criteria for the insolvability of x^2+D=y^n

Determination of class numbers for quadratic fields

Conditions for divisibility of class numbers

Abstract

In this paper we provide criteria for the insolvability of the Diophantine equation $x^2+D=y^n$. This result is then used to determine the class number of the quadratic field $\mathbb{Q}(\sqrt{-D})$. We also determine some criteria for the divisibility of the class number of the quadratic field $\mathbb{Q}(\sqrt{-D})$ and this result is then used to discuss the solvability of the Diophantine equation $x^2+D=y^n$.