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

Journal de Théorie des Nombres de Bordeaux (2013)

  • Volume: 25, Issue: 1, page 179-210
  • ISSN: 1246-7405

Abstract

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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 ∈ { 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 ≤ 3 except for a finite number of cases all of which satisfy max ( a , b , r , s , x , y ) < 2 · 10 15 for each solution; when gcd ( a , b ) > 1 , we show that N ≤ 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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  13. R. Scott, R. Styer, On p x - q y = c and related three term exponential Diophantine equations with prime bases. Journal of Number Theory 105 (2004), no. 2, 212–234. Zbl1080.11032
  14. R. Scott, R. Styer, On the generalized Pillai equation ± a x ± b y = c . Journal of Number Theory 118 (2006), 236–265. Zbl1146.11020
  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
  17. T. N. Shorey, On the equation a x m - b y n = k . Nederl. Akad. Wetensch. Indag. Math. 48 (1986), no. 3, 353–358. Zbl0603.10019
  18. R. Styer, Small two-variable exponential Diophantine equations. Mathematics of Computation 60 (1993), no. 202, 811–816. Zbl0781.11014
  19. M. Waldschmidt, Perfect powers: Pillai’s works and their developments. Arxiv preprint arXiv:0908.4031, August 27, 2009. 

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