Perfect powers with few binary digits and related Diophantine problems, II

Abstract

Abstract. We prove that, for any fixed base x ≥ 2 and sufficiently large prime q, no perfect q-th powers can be written with 3 or 4 digits 1 in base x. This is a particular instance of rather more general results, whose proofs follow from a combination of refined lower bounds for linear forms in Archimedean and non-Archimedean logarithms.

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

“…contradicting the main theorem of [11] (see the Theorem in the introduction of [11], noting that p > 2 is intended; see also Proposition 4.1 in the Appendix of the present paper) unless (a, b, c) = (3, 5, 2) or (5,11,4). Considering each of these two cases modulo 3, we see that neither case allows a solution to (1.2) with y odd, so neither case has a third solution.…”

“…( [16] has the same paper referenced in [4] as R. Scott, Elementary treatment of p a ± p b + 1 = x 2 .) We give a shorter proof of Szalay's result for the case v = 0, using a bound of Bauer and Bennett [1]; we also point out that the proof for the case v = 1 can be made elementary.…”

“…• 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)…”

Effective irrationality measures for real and p-adic roots of rational numbers close to 1, with an application to parametric families of Thue–Mahler equations

“…A parameter also called log E appeared for the first time in the p-adic setting in [8] and allows one to get better estimates when the rational numbers involved in the linear form are p-adically close to 1. Some applications of these refined estimates have been given in [8], a more spectacular one can be found in [2]. Here, we apply it to get explicit uniform, effective irrationality measures for p-adic n-th roots of certain rational numbers.…”

On the Diophantine equation 1+2a+xb=yn

“…We only mention the results of Bennett, Bugeaud and Mignotte [4,5], Corvaja and Zannier [13] and Bennett and Bugeaud [3] (see also the references in these papers). Beside deriving certain finiteness results, the authors solve several diophantine equations of the type (1) 1 + x a 1 + x b 2 = y n .…”