Abstract
Let \((U_n)_{n\ge 0}\) be a fixed linear recurrence sequence of integers with order at least two, and for any positive integer \(\ell \), let \(\ell \cdot 2^{\ell } + 1\) be a Cullen number. Recently in Bilu et al. (J Number Theory 202:412–425, 2019), generalized Cullen numbers in terms of linear recurrence sequence \((U_n)_{n\ge 0}\) under certain weak assumptions has been studied. In this paper, we consider the more general Diophantine equation \(U_{n_1} + \cdots + U_{n_k} = \ell \cdot x^{\ell } + Q(x)\), for a given polynomial \(Q(x) \in \mathbb {Z}[x]\) and prove an effective finiteness result. Furthermore, we demonstrate our method by an example.
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References
Bérczes, A., Hajdu, L., Pink, I., Rout, S.S.: Sums of \(S\)-units in recurrence sequences. J. Number Theory 196, 353–363 (2019)
Berrizbeitia, P., Fernandes, J.G., Gonzńskialez, M., Luca, F., Janitzio, V.: On Cullen numbers which are both Riesel and Sierpiński numbers. J. Number Theory 132, 2836–2841 (2012)
Bilu, Y., Marques, D., Togbé, A.: Generalized Cullen numbers in linear recurrence sequences. J. Number Theory 202, 412–425 (2019)
Bugeaud, Y., Mignotte, M., Siksek, S.: Classical and modular approaches to exponential Diophantine equations I. Fibonacci and Lucas perfect powers. Ann. Math. 163, 969–1018 (2006)
Cullen, J.: Question 15897 Educ. Times 534 (1905)
Dujella, A., Pethő, A.: A generalization of a theorem of Baker and Davenport. Quart. J. Math. Oxford Ser. 49, 291–306 (1998)
Guy, R.: Unsolved Problems in Number Theory. Springer, New York (1994)
Hooley, C.: Application of the Sieve Methods to the Theory of Numbers. Cambridge University Press, Cambridge (1976)
Lenstra, A.K., Lenstra, H.W., Jr., Lovász, L.: Factoring polynomials with rational coefficients. Math. Ann. 261(4), 515–534 (1982)
Luca, F., Shparlinski, I.: Pseudoprime Cullen and Woodall numbers. Colloq. Math. 107, 35–43 (2007)
Luca, F., Stănică, P.: Cullen numbers in binary recurrent sequences. In: Applications of Fibonacci Numbers, Vol. 10. Kluwer Academic Publishers, pp. 167–175 (2004)
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, 125–180 (2000). English translation. In Izv. Math. 64, 1217–1269 (2000)
Mazumdar, E., Rout, S.S.: Prime powers in sums of terms of binary recurrence sequences. Monatsh. Math. 189, 695–714 (2019)
Meher, N.K., Rout, S.S.: Linear combinations of prime powers in sums of terms of binary recurrence sequences. Lith. Math. J. 57(4), 1–15 (2017)
Smart, N.P.: The Algorithmic Resolution of Diophantine Equations, London Mathematical Society Student Texts, vol. 41. Cambridge University Press, Cambridge (1998)
Shorey, T.N., Tijdeman, R.: Exponential Diophantine Equations. Cambridge University Press, Cambridge (1986)
Stroeker, R., de Weger, B.M.M.: Solving elliptic Diophantine equations: the general cubic case. Acta Arith. 87(4), 339–365 (1999)
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The second author thanks the SERB for support under CRG grant CRG/2022/000268. The authors sincerely thank the referee for her/his thorough reviews and suggestions which significantly improves the quality of the paper.
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Meher, N.K., Rout, S.S. Cullen Numbers in Sums of Terms of Recurrence Sequence. Results Math 78, 102 (2023). https://doi.org/10.1007/s00025-023-01880-z
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DOI: https://doi.org/10.1007/s00025-023-01880-z