On some exponential equations of S. S. Pillai (Q2753429)

An old and yet unproved conjecture of Pillai states that for any given positive integer \(c\), there are only finitely many quadruples of integers \((a,x,b,y)\) with \(a^x-b^y=c\), \(a>1\), \(x>1\), \(b>1\), \(y>1\). The author considers a special case of this equation, namely (1) \(a^x-b^y=c\) in integers \(x>0\), \(y>0\), where \(a,b,c\) are given integers with \(a,b\geq 2\), \(c\geq 1\). Already in 1918, Polyá proved that (1) has only finitely many solutions, and Pillai proved that this equation has at most one solution if \(\text{gcd}(a,b)=1\) and \(c\) is sufficiently large. Pillai's result is ineffective, in that the required lower bound for \(c\) is not effectively computable. NEWLINENEWLINENEWLINEIn the present paper, the author proves that (1) has at most two solutions in positive integers \(x,y\), without any conditions imposed on \(a,b,c\) other than \(a,b\geq 2\), \(c\geq 1\). This improves upon earlier results of Le and Shorey who in the special case \(\text{gcd}(a,b)=1\), \(\min (a,b)\geq 10^5\) proved that (1) has at most two solutions with \(x>1,y>1\) (the author includes also solutions with \(x=1\) or \(y=1\)). The author's argument rests upon a sharp lower bound for linear forms in two logarithms due to Laurent, Mignotte and Nesterenko, as well as an elementary but ingenious gap argument. NEWLINENEWLINENEWLINEThe author conjectures that (1) has at most one solution, except for those triples \((a,b,c)\) belonging to some finite, explicit list given in his paper. He proves several special cases of this conjecture. First he proves his conjecture for triples \((a,b,c)\) with \(c\leq 100\). Second, he shows that (1) has at most one solution if \(c\geq b^{2a^2\log a}\); this is an effective version of Pillai's result mentioned above. The author proves also a counterpart, implying that (1) has at most one solution if \(c\) is sufficiently small in terms of \(a,b\); this extends a result of Terai, who proved a result of this shape if \(\text{gcd}(a,b)=1\) and \(a-b=c\).