Let {P n}n≥0 be the sequence of Padovan numbers defined by P0 = 0, P1 = 1, P2 = 1, and Pn+3 = Pn+1 + Pn for all n ≥ 0. In this paper, we find all integers c admitting at least two representations as a difference between a Padovan number and a power of 3.
Baker, A. and Davenport, H., The equations 3x 2 − 2 = y 2 and 8x 2 − 7 = z 2, Q. J. Math.., 20(1) (1969), 129–137,
Baker, A. and Wüstholz, G., Logarithmic forms and Diophantine geometry, vol. 9. Cambridge University Press, 2008.
Bugeaud, Y., Mignotte, M. and Siksek, S., Classical Classical and modular approaches to exponential Diophantine equations I. Fibonacci and Lucas perfect powers, Ann. of Math., 163(2) (2006), 969–1018.
Bravo, J. J., Luca, F. and Yazán, K., On Pillai's problem with Tribonacci numbers and powers of 2, Bull. Korean Math. Soc. 54(3) (2017), 1069–1080,
Chim, K. C., Pink, I. and Ziegler, V., On a variant of Pillai's problem, Int. J. Number Theory, 13(7) (2017), 1711–1727,
Chim, K. C., Pink, I. and Ziegler, V., On a variant of Pillai's problem II, J. Number Theory, 183 (2018), 269–290.
Ddamulira, M., On the problem of Pillai with Fibonacci numbers and powers of 3, Bol. Soc. Mat. Mex., To appear, 2019.
Ddamulira, M., On the problem of Pillai with Tribonacci numbers and powers of 3, J. Integer Seq., 22(5) (2019), Art. 19.5.6.
Ddamulira, M. and Luca, F., On a problem of Pillai with k-generalized Fibonacci numbers and powers of 3, Preprint, 2019.
Ddamülira, M., Lüca, F. and Rakotomalala, M., On a problem of Pillai with Fibonacci numbers and powers of 2, Proc. Indian Acad. Sci. Math. Sci., 127(3) (2017), 411–421.
Ddamulira, M., Gómez Ruiz, C. A. and Luca, F., On a problem of Pillai with k-generalized Fibonacci numbers and powers of 2, Monatsh. Math., 187(4) (2018), 635–664.
García Lomeli, A. M. and Hernández Hernández, S., Pillai's problem with Padovan numbers and powers of two, Rev. Colombiana Mat., 53(1) (2019), 1–14.
Hernane, M. O., Luca, F., Rihane, S. E. and Togbé, A., On Pillai's problem with Pell numbers and powers of 2, Hardy-Ramanujan J., 41 (2018), 22–31.
Hernandez, S. H., Luca, F. and Rivera, L. M. On Pillai's problem with the Fibonacci and Pell sequences, Bol. Soc. Mat. Mex., 2018.
Dujella, A. and Pethő, A., A generalization of a theorem of Baker and Davenport, Q. J. Math., 49(195) (1998), 291–306.
Gúzman, S. S. and Luca, F., Linear combinations of factorials and s-units in a binary recurrence sequence, Ann. Math. Qué., 38(2) (2014), 169–188.
Herschfeld, A., The equation 2x − 3y = d, Bull. Amer. Math. Soc., 41 (1935), 631.
Herschfeld, A., The equation 2x − 3y = d, Bull. Amer. Math. Soc., 42 (1936), 231–234.
Matveev, E. M. An explicit lower bound for a homogeneous rational linear form in the logarithms of algebraic numbers II, Izv. Ross. Akad. Nauk Ser. Mat. 64(6) (2000), 125–180, in Russian; English translation in Izv. Math. 64(6) (2000), 1217–1269.
Mihăilescu, P., Primary cyclotomic units and a proof of Catalan's conjecture, J. Reine Angew. Math., 572 (2006), 167–195.
OEIS Foundation Inc., The On-Line Encyclopedia of Integer Sequences, 2019, https://oeis.org.
Pillai, S. S. On ax −by = c, J. Indian Math. Soc. (N.S.), 2 (1936), 119–122.
Pillai, S. S. A correction to the paper On ax − by = c, J. Indian Math. Soc. (N.S.), 2 (1937), 215.