Abstract :
Ifaandbare distinct positive integers then a previous result of the author implies that the simultaneous Diophantine equationsx2−az2=y2−bz2=1possess at most 3 solutions in positive integers (x, y, z). On the other hand, there are infinite families of distinct integers (a, b) for which the above equations have at least 2 positive solutions. For each such family, we prove that there are precisely 2 solutions, with the possible exceptions of finitely many pairs (a, b). Since these families provide essentially the only pairs (a, b) for which the above equations are known to have more than a single solution (in positive (x, y, z)), this lends support to the conjecture that the number of such solutions to the above equations is less-than-or-equals, slant2 in all cases.
