Open AccessThue's equation as a tool to solve two different problems
Author(s) -
Sadek Bouroubi,
Ali Debbache
Publication year - 2021
Publication title -
acta et commentationes universitatis tartuensis de mathematica./acta et commentationes universitatis tartuensis de mathematica
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.276
H-Index - 6
eISSN - 2228-4699
pISSN - 1406-2283
DOI - 10.12697/acutm.2021.25.10
Subject(s) - diophantine equation , mathematics , thue equation , integer (computer science) , tuple , degree (music) , cube (algebra) , set (abstract data type) , product (mathematics) , combinatorics , discrete mathematics , binary number , diophantine set , arithmetic , computer science , physics , geometry , acoustics , programming language
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 D13-set if the product of any of its three distinct elements is a perfect cube minus one. We prove that any D13-set is finite and, for any positive integer a, the two-tuple {a, 2a} is extendible to a D13-set 3-tuple, but not to a 4-tuple. Using the well-known Thue equation 2x3 - y3 = 1, we show that the only cubic-triangular number is 1.