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

Reese Scott; Robert Styer

doi:10.5802/jtnb.902

Reese Scott ¹; Robert Styer ²

¹ 86 Boston St Somerville, MA 02143-2014, USA
² Villanova University 800 Lancaster Avenue Villanova, PA, USA

Journal de théorie des nombres de Bordeaux, Volume 27 (2015) no. 1, pp. 289-307.

Abstract
Résumé

Bennett has proven: If $a$ , $b$ , and $c$ are positive integers with $a, b \geq 2$ , then the equation $a^{x} - b^{y} = c$ has at most two solutions in positive integers $x$ and $y$ . Here we generalize this by allowing $a$ , $b$ , and $c$ to be positive rational numbers and, further, allowing $x$ and $y$ to be any integers, positive, negative, or zero. There are still at most two solutions except for two designated cases.

Bennett a démontré que si $a$ , $b$ et $c$ sont des nombres entiers positifs avec $a \geq 2$ et $b \geq 2$ , l’équation $a^{x} - b^{y} = c$ n’admet au plus que deux nombres entiers positifs $x$ et $y$ , comme solution. Nous pouvons généraliser ceci en choisissant $a$ , $b$ et $c$ dans l’ensemble des nombres rationnels positifs permettant à $x$ et $y$ d’être des nombres entiers, positifs, négatifs ou nuls. Il n’y a quand même, au plus, que deux solutions à l’exception de deux case où l’équation a exactement trois solutions.

MR: 3346973

DOI: 10.5802/jtnb.902

Classification: 11D61
Keywords: Pillai’s equation, Exponential Diophantine equations

Author's affiliations:

Reese Scott ¹; Robert Styer ²

¹ 86 Boston St Somerville, MA 02143-2014, USA
² Villanova University 800 Lancaster Avenue Villanova, PA, USA

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     title = {Bennett{\textquoteright}s {Pillai} theorem with fractional bases and negative exponents allowed},
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     publisher = {Soci\'et\'e Arithm\'etique de Bordeaux},
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Reese Scott; Robert Styer. Bennett’s Pillai theorem with fractional bases and negative exponents allowed. Journal de théorie des nombres de Bordeaux, Volume 27 (2015) no. 1, pp. 289-307. doi : 10.5802/jtnb.902. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.902/

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