On Some Exponential Equations of S. S. Pillai

Abstract

Abstract. In this paper, we establish a number of theorems on the classic Diophantine equation of S. S. Pillai, a x − b y = c, where a, b and c are given nonzero integers with a, b ≥ 2. In particular, we obtain the sharp result that there are at most two solutions in positive integers x and y and deduce a variety of explicit conditions under which there exists at most a single such solution. These improve or generalize prior work of Le, Leveque, Pillai, Scott and Terai. The main tools used include lower bounds for linear forms in the logarithms of (two) algebraic numbers and various elementary arguments.

Bennett’s Pillai theorem with fractional bases and negative exponents allowed

“…In [3], Bennett handles Theorem A for the case a prime by using results from [14], but this will not work here since we are allowing the exponents x and y to be zero. Instead, we will use a result of Luca [11] on the equation p r ± p s + 1 = z 2 , where p, r, s, and z are positive integers with p prime.…”

“…Before using lower bounds on linear forms in logarithms in the proof of Theorem 1, we need to justify the use of a ≥ 6 as in [3] by first proving Theorem 1 for the case a prime. In [3], Bennett handles Theorem A for the case a prime by using results from [14], but this will not work here since we are allowing the exponents x and y to be zero.…”

“…• if max(x 2 , y 1 , y 2 ) = 4, then min(x 2 , max(y 1 , y 2 )) < 3, and let S 2 = {(1, 1, 4, 4), (2,2,4,4), (3,3,4,4), (3,4,4,3)…”

“…In [3], Bennett proves Reese Scott, Robert Styer Equation (1.1) of Theorem A is generally known as the Pillai equation; brief histories are given in [3] and [14], but see [18] for a much more extended history.…”

“…The first step towards these generalizations is the following: has at most two solutions in nonnegative integers x and y except for (a, b, c) = (2, 5, 3), which has solutions (x, y) = (2, 0), (3,1), (7,3), and no further solutions.…”