On the Equations px − by = c and ax + by = cz

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.…”

“…We apply Theorem 3 of [14] which states that, when a is prime and the parity of y is fixed, Equation (1.1) has at most one solution in positive integers (x, y) except for the following five exceptional (a, b, c; x 1 , y 1 , x 2 , y 2 ): (3, 2, 1; 1, 1, 2, 3), (2, 3, 5; 3, 1, 5, 3), (2, 3, 13; 4, 1, 8, 5), (2, 5, 3; 3, 1, 7, 3), (13, 3, 10; 1, 1, 3, 7). Noting that none of these five exceptional cases has a further solution with 2 | y > 0 (use congruences modulo 3 or modulo 8), we see that, if (1.2) has more than two solutions in nonnegative integers x and y when a is prime, we must have exactly one solution with y = 0 and exactly two further solutions.…”

“…If these two further solutions are among the five exceptional cases, a solution with y = 0 occurs only when (a, b, c) = (2,5,3), in which case the three solutions are (2, 0), (3, 1), (7,3). So from here on we exclude the five exceptional cases of Theorem 3 of [14] so that we can assume that we have three solutions (x 1 , y 1 ), (x 2 , y 2 ), (x 3 , y 3 ) with y 1 = 0 and 2 | y 2 − y 3 . Without loss of generality, assume 2 | y 2 = 2t for some integer t. Then we have a solution to the equation…”

“…Now we apply Theorem 4 of [14] which states that, if a = 2, Equation (1.1) has at most one solution in positive integers x and y except when (a, b, c; x 1 , y 1 , x 2 , y 2 ) is one of (2, 3, 5; 3, 1, 5, 3), (2, 3, 13; 4, 1, 8, 5), (2, 5, 3; 3, 1, 7, 3). Applying this result to the solutions (x 2 , y 2 ) and (x 3 , y 3 ) and noting that all three cases just listed have already been excluded, we see that a must be an odd prime.…”

“…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.…”