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On the Diophantine equation \(F_n + F_m=2^a\).(English)Zbl 1419.11024

Summary: In this paper, we find all the solutions of the title Diophantine equation in positive integer variables \((n,m,a)\), where \(F_k\) is the \(k\)th term of the Fibonacci sequence. The proof of our main theorem uses lower bounds for linear forms in logarithms (Baker’s theory) and a version of the Baker-Davenport reduction method in diophantine approximation.

MSC:

11B39 Fibonacci and Lucas numbers and polynomials and generalizations
11J86 Linear forms in logarithms; Baker’s method
11D61 Exponential Diophantine equations

Cite

References:

[1]Baker, A.; Davenport, H., The equations \(3x^2\)−\(2 = y^2\) and \(8x^2\)−\(7 = z^2\), Quart.J. Math. Oxford Ser. (2), 20, 1, 129-137 (1969) ·Zbl 0177.06802
[2]Bilu, Yu.; Hanrot, G.; Voutier, P., Existence of primitive divisors of Lucas and Lehmer numbers (with an appendix by M. Mignotte, J. Reine Angew. Math., 539, 75-122 (2001) ·Zbl 0995.11010
[3]Bravo, J. J.; Luca, F., On a conjecture about repdigits in k−generalized Fi- bonacci sequences, Publ. Math. Debrecen, 82, 3-4, 623-639 (2013) ·Zbl 1274.11035
[4]Bugeaud, Y.; Mignotte, M.; Siksek, S., Classical and modular approaches to exponential Diophantine equations. I. Fibonacci and Lucas perfect powers, Ann. of Math. (2), 163, 3, 969-1018 (2006) ·Zbl 1113.11021
[5]Carmichael, R. D., On the numerical factors of the arithmetic forms\(α^n\) ± \(β^n\), Ann. Math., 15, 1-4, 30-70 (1913) ·JFM 44.0216.01
[6]D´Iaz Alvarado, S.; Luca, F.; Luca, F.; Stanica, P., Fibonacci numbers which are sums of two repdigits, Proceedings of the XIVth International Conference on Fibonacci numbers and their applications, 97-111 (2011) ·Zbl 1287.11021
[7]Dujella, A.; Petho˝, A., A generalization of a theorem of Baker and Davenport, Quart. J. Math. Oxford Ser. (2), 49, 3, 291-306 (1998) ·Zbl 0911.11018
[8]Koshy, T., Fibonacci and Lucas Numbers with Applications (2001), Wiley-Interscience Pub- lication: Wiley-Interscience Pub- lication, New York ·Zbl 0984.11010
[9]Luca, F., Repdigits as sums of three Fibonacci numbers, Math. Commun., 17, 1-11 (2012) ·Zbl 1305.11008
[10]Luca, F.; Siksek, S., On factorials expressible as sums of at most three Fibonacci numbers, Proc. Edinb. Math. Soc. (2), 53, 3, 747-763 (2010) ·Zbl 1253.11048
[11]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, 125-180 (2000) ·Zbl 1013.11043
[12]Petho˝, A.; Tichy, R. F., S-unit equations, linear recurrences and digit expansions, Publ. Math. Debrecen, 42, 1-2, 145-154 (1993) ·Zbl 0792.11006
[13]Senge, H. G.; Straus, E. G., PV-numbers and sets of multiplicity, Period. Math. Hungar., 3, 93-100 (1973) ·Zbl 0248.12004
[14]Stewart, C. L., On the representation of an integer in two different bases, J. Reine Angew. Math., 319, 63-72 (1980) ·Zbl 0426.10008
[15]Zeckendorf, E., Repr´esentation des nombres naturels par une somme de nombres de Fibonacci ou de nombres de Lucas, Bull. Soc. Roy. Sci. Li‘ege, 41, 179-182 (1972) ·Zbl 0252.10011
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
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