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

Two terms with known prime divisors adding to a power

“…If ǫ = 1, then x 1 = 1, and we have the infinite family which is the final exception in the formulation of Theorem 2. Note that members of this exceptional infinite family are the only cases in which (1.1) has more than one solution with min(X, Y ) = 1, by Lemma 3.2 of [15].…”

“…Thus, (5.3) and (5.4) are the only solutions to (5.2) such that XY = d k1 2 k2 where k 1 is odd and k 2 − (g − 1) is even. Also, recall (5.2) has no further solutions with min(X, Y ) = 1, by Lemma 3.2 of [15].…”

“…For Case 1 we have 3 ν −1 = 2c j , so that, by Lemmas 3.1 and 3.2 of [15], c determines ν which determines A and B, so that a unique {c, c} ∈ K is determined. And if (A, B, z) is a Case 2 solution associated with a given {c, c} ∈ K, then 1 + 4AB = c 2z is a solution to (2.2) associated with the same {c, c}, so that, by Lemmas 3.1 and 3.2 of [15], a unique {c, c} ∈ K is determined.…”

“…, which means we have a solution (A, B, j) with |A − B| = 1, thus we have Case 2. Any further solutions with 2j | z > 2j associated with {c, c} must have A = 1; by Lemmas 3.1 and 3.2 of [15], this is impossible.…”

An ancient diophantine equation with applications to numerical curios and geometric series

“…The solution of (6) found by Fermat is given below: by Scott and Styer [4] who, referring to Skolem's book [5], have stated that this equation has infinitely many solutions in positive rational numbers.…”