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- W4200195962 abstract "Let a,b,k be fixed positive integers such that min{a,b}>1 and gcd(a,b)=1, and let N′(a,b,k) denote the number of positive integer solutions (x,y) of the equation |ax−by|=k. In this paper, using a lower bound of linear forms in two logarithms combined with some properties of convergents of irrational numbers, we prove the following two results: (i) If min{a,b}≥85988, then N′(a,b,k)≤2. (ii) For any real number ε with 0<ε<1, if k<min{a1−ε,b1−ε} and max{a,b}>C(ε), where C(ε) is an effectively computable constant depending only on ε, then N′(a,b,k)≤1. In particular, if k<min{a1/15,b1/15}, then N′(a,b,k)≤1 except for N′(2,3,1)=N′(3,2,1)=3." @default.
- W4200195962 created "2021-12-31" @default.
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- W4200195962 date "2022-10-01" @default.
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- W4200195962 title "A note on the number of solutions of the Pillai type equation <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML altimg=si1.svg><mml:mrow><mml:mo stretchy=true>|</mml:mo><mml:msup><mml:mrow><mml:mi>a</mml:mi></mml:mrow><mml:mrow><mml:mi>x</mml:mi></mml:mrow></mml:msup><mml:mo linebreak=goodbreak linebreakstyle=after>−</mml:mo><mml:msup><mml:mrow><mml:mi>b</mml:mi></mml:mrow><mml:mrow><mml:mi>y</mml:mi></mml:mrow></mml:msup><mml:mo stretchy=true>|</mml:mo></mml:mrow><mml:mo …" @default.
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- W4200195962 doi "https://doi.org/10.1016/j.jnt.2021.12.001" @default.
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