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.

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DOI : https://doi.org/10.5802/jtnb.832
Classification : 11D61
Mots clés : Pillai’s equation, Exponential Diophantine equations
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     author = {Reese Scott and Robert Styer},
     title = {The number of solutions to the generalized {Pillai} equation $\pm r a^x \pm s b^y = c$.},
     journal = {Journal de Th\'eorie des Nombres de Bordeaux},
     pages = {179--210},
     publisher = {Soci\'et\'e Arithm\'etique de Bordeaux},
     volume = {25},
     number = {1},
     year = {2013},
     doi = {10.5802/jtnb.832},
     zbl = {1271.11038},
     mrnumber = {3063837},
     language = {en},
     url = {https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.832/}
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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/

[1] M. Bennett, On some exponential equations of S. S. Pillai. Canadian Journal of Mathematics 53 (2001) no. 5, 897–922.

[2] Y. F. Bilu, Y. Bugeaud, M. Mignotte, Catalan’s Equation, book in preparation.

[3] Y. Bugeaud, F. Luca, On Pillai’s Diophantine equation. New York J. of Math. 12 (2006), 193–217 (electronic).

[4] R. K. Guy, C. B. Lacampagne, J. L. Selfridge, Primes at a glance. Math. Comp. 48 (1987), 183–202.

[5] B. He, A. Togbé, On the number of solutions of the exponential Diophantine equation ax m -by n =c. Bull. Aust. Math. Soc. 81 (2010), 177–185.

[6] M. Le, A note on the diophantine equation ax m -by n =k. Indag. Math. (N. S.) 3 (June 1992), 185–191.

[7] E. M. Matveev, An explicit lower bound for a homogeneous rational linear form in logarithms of algebraic numbers. II. Izv. Math. 64 (2000), 1217–1269.

[8] M. Mignotte, A corollary to a theorem of Laurent-Mignotte-Nesterenko. Acta Arith. 86 (1998), 101–111.

[9] M. Mignotte, A kit on linear forms in three logarithms, in preparation, February 7, 2008.

[10] S. S. Pillai, On the inequality 0<a x -b y n. J. Indian Math. Soc. (1) 19 (1931), 1–11.

[11] S. S. Pillai, On the equation 2 x -3 y =2 X +3 Y . Bull. Calcutta Soc. 37 (1945), 15–20.

[12] R. Scott, On the Equations p x -b y =c and a x +b y =c z . Journal of Number Theory 44 (1993), no. 2, 153–165.

[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.

[14] R. Scott, R. Styer, On the generalized Pillai equation ±a x ±b y =c. Journal of Number Theory 118 (2006), 236–265.

[15] R. Scott, R. Styer, The generalized Pillai equation ±ra x ±sb y =c. Journal of Number Theory 131 (2011), 1037–1047.

[16] R. Scott, R. Styer, Handling a large bound for a problem on the generalized Pillai equation ±ra x ±sb y =c. Preprint.

[17] T. N. Shorey, On the equation ax m -by n =k. Nederl. Akad. Wetensch. Indag. Math. 48 (1986), no. 3, 353–358.

[18] R. Styer, Small two-variable exponential Diophantine equations. Mathematics of Computation 60 (1993), no. 202, 811–816.

[19] M. Waldschmidt, Perfect powers: Pillai’s works and their developments. Arxiv preprint arXiv:0908.4031, August 27, 2009.