The number of solutions to the generalized Pillai equation ±ra x ±sb y =c.
Journal de théorie des nombres de Bordeaux, Tome 25 (2013) no. 1, pp. 179-210.

Nous considérons N, le nombre de solutions (x,y,u,v) de l’équation (-1) u ra x +(-1) v sb y =c en nombres entiers non négatifs x, y, et nombres entiers u,v{0,1}, pour des entiers donnés a>1, b>1, c>0, r>0 et s>0. Lorsque pgcd(ra,sb)=1, nous montrons que N3, sauf pour un nombre fini de cas qui satisfont à max(a,b,r,s,x,y)<2·10 15 pour chaque solution ; lorsque pgcd(a,b)>1, nous montrons que N3 sauf pour trois familles infinies de cas exceptionnels. Nous trouvons plusieurs façons de générer un nombre infini de cas donnant N=3 solutions.

We consider N, the number of solutions (x,y,u,v) to the equation (-1) u ra x +(-1) v sb 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(ra,sb)=1, we show that N3 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 N3 except for three infinite families of exceptional cases. We find several different ways to generate an infinite number of cases giving N=3 solutions.

DOI : 10.5802/jtnb.832
Classification : 11D61
Mots clés : Pillai’s equation, Exponential Diophantine equations
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, Tome 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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