The number of solutions to the generalized Pillai equation $\pm r a^x \pm s b^y = c$.

Reese Scott; Robert Styer

doi:10.5802/jtnb.832

The number of solutions to the generalized Pillai equation

\pm r a^{x} \pm s b^{y} = c

.

Reese Scott ¹; Robert Styer ²

¹ Somerville, MA, USA
² Villanova University 800 Lancaster Avenue Villanova, PA, USA

Journal de théorie des nombres de Bordeaux, Volume 25 (2013) no. 1, pp. 179-210.

Abstract
Résumé

We consider $N$ , the number of solutions $(x, y, u, v)$ to the equation ${(- 1)}^{u} r a^{x} + {(- 1)}^{v} s b^{y} = c$ in nonnegative integers $x, y$ and integers $u, v \in {0, 1}$ , for given integers $a > 1$ , $b > 1$ , $c > 0$ , $r > 0$ and $s > 0$ . When $gcd (r a, s b) = 1$ , we show that $N \leq 3$ except for a finite number of cases all of which satisfy $max (a, b, r, s, x, y) < 2 \cdot 10^{15}$ for each solution; when $gcd (a, b) > 1$ , we show that $N \leq 3$ except for three infinite families of exceptional cases. We find several different ways to generate an infinite number of cases giving $N = 3$ solutions.

Nous considérons $N$ , le nombre de solutions $(x, y, u, v)$ de l’équation ${(- 1)}^{u} r a^{x} + {(- 1)}^{v} s b^{y} = c$ en nombres entiers non négatifs $x$ , $y$ , et nombres entiers $u, v \in {0, 1}$ , pour des entiers donnés $a > 1$ , $b > 1$ , $c > 0$ , $r > 0$ et $s > 0$ . Lorsque $pgcd (r a, s b) = 1$ , nous montrons que $N \leq 3$ , sauf pour un nombre fini de cas qui satisfont à $max (a, b, r, s, x, y) < 2 \cdot 10^{15}$ pour chaque solution ; lorsque $pgcd (a, b) > 1$ , nous montrons que $N \leq 3$ sauf pour trois familles infinies de cas exceptionnels. Nous trouvons plusieurs façons de générer un nombre infini de cas donnant $N = 3$ solutions.

MR Zbl

DOI: 10.5802/jtnb.832

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

Author's affiliations:

Reese Scott ¹; Robert Styer ²

¹ Somerville, MA, USA
² Villanova University 800 Lancaster Avenue Villanova, PA, USA

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Reese Scott; Robert Styer. The number of solutions to the generalized Pillai equation $\pm r a^x \pm s b^y = c$.. Journal de théorie des nombres de Bordeaux, Volume 25 (2013) no. 1, pp. 179-210. doi : 10.5802/jtnb.832. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.832/

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