[ Résolution et application de l’équation diophantienne AXY + BX + CY = D (A et BC étrangers) ]
Mushiwalyahyage Zaluka1, Déborah Amani Faraja2, and Marie-Chantal Niyomanzi Sebutimbiri3
1 Chef de Travaux, Département de Mathématique-Physique, Institut Supérieur Pédagogique de Bukavu (ISP, BUKAVU), Bukavu, RD Congo
2 Assistante, Ecole des Mines, Université Officielle de Bukavu (U.O.B), Bukavu, RD Congo
3 Assistante, Institut Supérieur des Techniques médicales de Goma (ISTM, GOMA), Goma, RD Congo
Original language: French
Copyright © 2026 ISSR Journals. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
This paper deals with the Diophantine equation axy+bx+cy=d (and bc are coprime). It revisits the old methods of solving this equation. It establishes a link between this solution and the characterization of the elements of the arithmetic progression ax+b (and b are coprime). It provides a new method for solving this equation. It leads to two primality criteria and a commutative diagram characterizing odd natural numbers.
Author Keywords: diophantine equation, coprime numbers, primality, commutative diagram, characterization.
Mushiwalyahyage Zaluka1, Déborah Amani Faraja2, and Marie-Chantal Niyomanzi Sebutimbiri3
1 Chef de Travaux, Département de Mathématique-Physique, Institut Supérieur Pédagogique de Bukavu (ISP, BUKAVU), Bukavu, RD Congo
2 Assistante, Ecole des Mines, Université Officielle de Bukavu (U.O.B), Bukavu, RD Congo
3 Assistante, Institut Supérieur des Techniques médicales de Goma (ISTM, GOMA), Goma, RD Congo
Original language: French
Copyright © 2026 ISSR Journals. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Abstract
This paper deals with the Diophantine equation axy+bx+cy=d (and bc are coprime). It revisits the old methods of solving this equation. It establishes a link between this solution and the characterization of the elements of the arithmetic progression ax+b (and b are coprime). It provides a new method for solving this equation. It leads to two primality criteria and a commutative diagram characterizing odd natural numbers.
Author Keywords: diophantine equation, coprime numbers, primality, commutative diagram, characterization.
Abstract: (french)
Cet article porte sur l’équation diophantienne axy+bx+cy=d (a et bc étrangers). Il en repense les anciennes méthodes de résolution. Il établit un lien entre cette résolution et la caractérisation des éléments de la progression arithmétique ax+b (a et b étrangers). Il en donne une nouvelle méthode de résolution. Il aboutit à deux critères de primalité et un diagramme commutatif de caractérisation des nombres entiers naturels impairs.
Author Keywords: équation diophantienne, nombres étrangers, primalité, diagramme commutatif, caractérisation.
