Bennett’s Pillai theorem with fractional bases and negative exponents allowed
Journal de théorie des nombres de Bordeaux, Tome 27 (2015) no. 1, pp. 289-307.

Bennett a démontré que si a, b et c sont des nombres entiers positifs avec a2 et b2, 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.

Bennett has proven: If a, b, and c are positive integers with a,b2, 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.

DOI : 10.5802/jtnb.902
Classification : 11D61
Mots clés : Pillai’s equation, Exponential Diophantine equations
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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Reese Scott; Robert Styer. Bennett’s Pillai theorem with fractional bases and negative exponents allowed. Journal de théorie des nombres de Bordeaux, Tome 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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