Pillai’s conjecture revisited
نویسنده
چکیده
We prove a generalization of an old conjecture of Pillai (now a theorem of Stroeker and Tijdeman) to the effect that the Diophantine equation 3 2 1⁄4 c has, for jcj > 13; at most one solution in positive integers x and y: In fact, we show that if N and c are positive integers with NX2; then the equation jðN þ 1Þ Nj 1⁄4 c has at most one solution in positive integers x and y; unless ðN; cÞAfð2; 1Þ; ð2; 5Þ; ð2; 7Þ; ð2; 13Þ; ð2; 23Þ; ð3; 13Þg: Our proof uses the hypergeometric method of Thue and Siegel and avoids application of lower bounds for linear forms in logarithms of algebraic numbers. r 2002 Elsevier Science (USA). All rights reserved. MSC: primary 11D61; 11D45
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Perfect Powers: Pillai’s Works and Their Developments
A perfect power is a positive integer of the form ax where a ≥ 1 and x ≥ 2 are rational integers. Subbayya Sivasankaranarayana Pillai wrote several papers on these numbers. In 1936 and again in 1945 he suggested that for any given k ≥ 1, the number of positive integer solutions (a, b, x, y), with x ≥ 2 and y ≥ 2, to the Diophantine equation ax − by = k is finite. This conjecture amounts to sayi...
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