Proof of a special case of Pillai’s conjecture

In the last post we conjectured that given fixed integers A, B, C, D, E>0 with BD>1, the equation

(*)\qquad\qquad\qquad\qquad\quad A\cdot B^x-C\cdot D^y=E

has only finitely many solutions (x,y) in integers. Let us prove this conjecture.

If (*) has no solution we are done. Otherwise suppose that (x_0,y_0) is a solution. Assume the contrary, and let S denote the infinite set of solutions (x,y). For any (x,y)\in S, we have

AB^x-AB^{x_0}=CD^y-CD^{y_0}.

So, if (*) has infinitely many solutions, then writing

\displaystyle u_x=AB^x-AB^{x_0},\quad v_y=CD^y-CD^{y_0}

gives that u_x=v_y for infinitely many x,y\in\mathbb N_0. So by Theorem 1.1 in this paper (or this one),

B^r=D^s

for some positive integers r,s. So \min\{B,D\}\mid\max\{B,D\} (exercise). But then \min\{B,D\}^{\min\{x,y\}} divides E for all (x,y)\in S, implying \min\{x,y\} is bounded over all (x,y)\in S. But then, by (*), \max\{x,y\} must also be bounded over all (x,y)\in S. Thus S is finite, a contradiction.

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