Triads of integers with equal sums of squares and equal products and a related multigrade chain

Ajai Choudhry ¹   ; Jarosław Wróblewski ²

¹ 13/4 A Clay Square Lucknow 226001, India
² Institute of Mathematics Wrocław University Pl. Grunwaldzki 2/4 50-384 Wrocław, Poland

Acta Arithmetica, Tome 178 (2017) no. 1, pp. 87-100

Résumé

While the complete solution of the problem of finding two triads of integers with equal sums of squares and equal products is known, till now no examples of three such triads has been published. In this paper we find several triads of positive integers with equal sums of squares and equal products. We present several parametric solutions for three such triads and give numerical examples for five triads of integers with equal sums of squares and equal products. We use these triads to produce multigrade chains of quadruples of integers with equal sums of $k$th powers for $k=1,2,3,5$. We conclude the paper with some open problems concerning such triads.

Détail

DOI : 10.4064/aa8574-9-2016

Keywords: while complete solution problem finding triads integers equal sums squares equal products known examples three triads has published paper several triads positive integers equal sums squares equal products present several parametric solutions three triads numerical examples five triads integers equal sums squares equal products these triads produce multigrade chains quadruples integers equal sums kth powers conclude paper problems concerning triads