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.

Reçu le : 2013-07-31
Accepté le : 2014-01-01
Publié le : 2015-05-21
DOI : https://doi.org/10.5802/jtnb.902
Classification : 11D61
Mots clés: Pillai’s equation, Exponential Diophantine equations
@article{JTNB_2015__27_1_289_0,
     author = {Reese Scott and Robert Styer},
     title = {Bennett's Pillai theorem with fractional bases and negative exponents allowed},
     journal = {Journal de Th\'eorie des Nombres de Bordeaux},
     publisher = {Soci\'et\'e Arithm\'etique de Bordeaux},
     volume = {27},
     number = {1},
     year = {2015},
     pages = {289-307},
     doi = {10.5802/jtnb.902},
     mrnumber = {3346973},
     language = {en},
     url = {jtnb.centre-mersenne.org/item/JTNB_2015__27_1_289_0/}
}
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/item/JTNB_2015__27_1_289_0/

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