On the Diophantine Pair {a,3a}

Abstract The purpose of the present paper is to consider the extensibility of the Diophantine triple { a , b , c } , where a b c with b = 3 a , and to prove that such a set cannot be extended to an irregular Diophantine quadruple. We succeed in that for some families of c's (depending on a). As corollary, we prove that any Diophantine quadruple which contains the pair { a , 3 a } is regular. Finally in this paper, we will see that by considering the case b = 8 a we obviously obtain similar results.