# The number of solutions to the generalized Pillai equation $±r{a}^{x}±s{b}^{y}=c$.

Reese Scott[1]; Robert Styer[2]

• [1] Somerville, MA, USA
• [2] Villanova University 800 Lancaster Avenue Villanova, PA, USA
• Volume: 25, Issue: 1, page 179-210
• ISSN: 1246-7405

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## Abstract

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

## How to cite

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Scott, Reese, and Styer, Robert. "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 25.1 (2013): 179-210. <http://eudml.org/doc/275700>.

@article{Scott2013,
abstract = {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 $u, v \in \lbrace 0,1\rbrace$, for given integers $a&gt;1$, $b&gt;1$, $c&gt;0$, $r&gt;0$ and $s&gt;0$. When $\gcd (ra,sb)=1$, we show that $N \le 3$ except for a finite number of cases all of which satisfy $\max (a,b,r,s, x,y) &lt; 2 \cdot 10^\{15\}$ for each solution; when $\gcd (a,b)&gt;1$, we show that $N \le 3$ except for three infinite families of exceptional cases. We find several different ways to generate an infinite number of cases giving $N=3$ solutions.},
affiliation = {Somerville, MA, USA; Villanova University 800 Lancaster Avenue Villanova, PA, USA},
author = {Scott, Reese, Styer, Robert},
journal = {Journal de Théorie des Nombres de Bordeaux},
keywords = {Pillai’s equation; Exponential Diophantine equations; Pillai's equation; exponential Diophantine equations},
language = {eng},
month = {4},
number = {1},
pages = {179-210},
publisher = {Société Arithmétique de Bordeaux},
title = {The number of solutions to the generalized Pillai equation $\pm r a^x \pm s b^y = c$.},
url = {http://eudml.org/doc/275700},
volume = {25},
year = {2013},
}

TY - JOUR
AU - Scott, Reese
AU - Styer, Robert
TI - The number of solutions to the generalized Pillai equation $\pm r a^x \pm s b^y = c$.
JO - Journal de Théorie des Nombres de Bordeaux
DA - 2013/4//
PB - Société Arithmétique de Bordeaux
VL - 25
IS - 1
SP - 179
EP - 210
AB - 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 $u, v \in \lbrace 0,1\rbrace$, for given integers $a&gt;1$, $b&gt;1$, $c&gt;0$, $r&gt;0$ and $s&gt;0$. When $\gcd (ra,sb)=1$, we show that $N \le 3$ except for a finite number of cases all of which satisfy $\max (a,b,r,s, x,y) &lt; 2 \cdot 10^{15}$ for each solution; when $\gcd (a,b)&gt;1$, we show that $N \le 3$ except for three infinite families of exceptional cases. We find several different ways to generate an infinite number of cases giving $N=3$ solutions.
LA - eng
KW - Pillai’s equation; Exponential Diophantine equations; Pillai's equation; exponential Diophantine equations
UR - http://eudml.org/doc/275700
ER -

## References

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15. R. Scott, R. Styer, The generalized Pillai equation $±r{a}^{x}±s{b}^{y}=c$. Journal of Number Theory 131 (2011), 1037–1047. Zbl1244.11030
16. R. Scott, R. Styer, Handling a large bound for a problem on the generalized Pillai equation $±r{a}^{x}±s{b}^{y}=c$. Preprint. Zbl1308.11040
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19. M. Waldschmidt, Perfect powers: Pillai’s works and their developments. Arxiv preprint arXiv:0908.4031, August 27, 2009.

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