Two conjectures on integer arithmetic and their applications to Diophantine equations

TL;DR

This paper discussed two conjectures related to integer arithmetic and their implications for Diophantine equations, but was withdrawn due to the falsity of Conjecture 1, with further details available online.

Contribution

The paper proposed two conjectures on integer arithmetic and explored their applications to Diophantine equations, but the primary conjecture was proven false.

Findings

Conjecture 1 is false.

Main results are available online.

Further analysis is documented at arXiv:0901.2093.

Abstract

This paper has been withdrawn by the author because Conjecture 1 is false. Please see arXiv:0901.2093 for a justification that Conjecture 1 is false. The other main results are also available from the above URL.