On Pillai's Diophantine equation (Q851550)

Let \(A\), \(B\), \(a\), \(b\) and \(c\) be fixed positive integers. The authors prove results on the number of solutions to Pillai's Diophantine equation \(Aa^x-Bb^y=c\) in positive unknown integers \(x\) and \(y\). Their first result is Theorem 1. Let \({\mathcal S}\) be the set of nonzero integers whose prime factors belong to some fixed finite set of prime numbers. Let also \(b\) be a fixed nonzero integer. Then the Diophantine equation \[ A(a^{x_1}-a^{x_2})=B (b^{y_1}-b^{y_2}) \] in positive unknown integers \(A\), \(B\), \(a\), \(x_1\), \(x_2\), \(y_1\), \(y_2\) has only finitely many solutions with \(x_1 \not= x_2\), \(A\), \(B\in {\mathcal S}\) and \(\gcd(aA,bB)=1\). The proof of this result uses applications of (Archimedian and non-Archimedian) linear forms in logarithms, Ridout's theorem and the general Subspace Theorem. A second result is Theorem 2. The ABC conjecture implies that the Diophantine equation \[ a^{x_1}-a^{x_2}=b^{y_1}-b^{y_2} \] has only finitely many positive integer solutions \((a, b, x_1, x_2, y_1, y_2)\) with \(a>1\), \(b>1\), \(x_1 \not= x_2\) and \(a^{x_1}\not=b^{y_1}\). To prove Theorem 2 they also prove an intermediate unconditional result Theorem 3. Assume that \(x_1> x_2>0\) and \(y_1> y_2>0\) are fixed integers with \(\gcd(x_1,x_2)=\gcd(y_1,y_2)=1\) and \(y_1>x_1\). Then the Diophantine equation \[ a^{x_1}-a^{x_2}=b^{y_1}-b^{y_2} \] has only finitely many positive integer solutions \((a, b)\).