Explorations in Diophantine Equations

Diophantine equations are equations containing only sums, products and powers where all constants are integers and the only solution of interest is an integer. The congruence method provides a convenient tool for determining the number of solutions to Diophantine equations. We locate the triple involving several special numbers in such a way that the triple whose product of any two of its parts is a perfect square. In this way, various triples and quadruples with different attributes can be found.