# Thue's equation as a tool to solve two different problems

Keywords:
D_1^3-set, Diophantine equation, cubic-triangular number, Thue's equation

### Abstract

A Thue equation is a Diophantine equation of the form *f*(*x; y*) = *r*, where *f *is an irreducible binary form of degree at least 3, and *r *is a given nonzero rational number. A set *S *of at least three positive integers is called a *D*1^{3}-set if the product of any of its three distinct elements is a perfect cube minus one. We prove that any *D*1^{3}-set is finite and, for any positive integer *a*, the two-tuple *{a, *2*a} *is extendible to a *D*1^{3}-set 3-tuple, but not to a 4-tuple. Using the well-known Thue equation 2*x ^{3}*

*- y*

^{3 }= 1, we show that the only cubic-triangular number is 1.

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