Fibonacci numbers that are not sums of two prime powers

Luca, Florian; Stănică, Pantelimon

doi:10.1090/S0002-9939-05-07827-5

Fibonacci numbers that are not sums of two prime powers
HTML articles powered by AMS MathViewer

by Florian Luca and Pantelimon Stănică PDF

Proc. Amer. Math. Soc. 133 (2005), 1887-1890 Request permission

Abstract:

In this paper, we construct an infinite arithmetic progression $\mathcal A$ of positive integers $n$ such that if $n\in {\mathcal A}$, then the $n$th Fibonacci number is not a sum of two prime powers.

References

Fred Cohen and J. L. Selfridge, Not every number is the sum or difference of two prime powers, Math. Comp. 29 (1975), 79–81. MR 376583, DOI 10.1090/S0025-5718-1975-0376583-0
J. G. van der Corput, On de Polignac’s conjecture, Simon Stevin 27 (1950), 99–105 (Dutch). MR 35298
P. Erdös, On integers of the form $2^k+p$ and some related problems, Summa Brasil. Math. 2 (1950), 113–123. MR 44558
J.-H. Evertse, K. Győry, C. L. Stewart, and R. Tijdeman, $S$-unit equations and their applications, New advances in transcendence theory (Durham, 1986) Cambridge Univ. Press, Cambridge, 1988, pp. 110–174. MR 971998
Richard K. Guy, Unsolved problems in number theory, 2nd ed., Problem Books in Mathematics, Springer-Verlag, New York, 1994. Unsolved Problems in Intuitive Mathematics, I. MR 1299330, DOI 10.1007/978-1-4899-3585-4
Florian Luca, On integers which are not differences of two powers, Mathematika 49 (2002), no. 1-2, 231–243 (2004). MR 2059057, DOI 10.1112/S002557930001620X
A. de Polignac, ‘Recherches nouvelles sur les nombres premiers’, C.R. Acad. Sci. Paris Math. 29 (1949), 397–401, 738–739.
N. P. Romanoff, ‘Über einige Sätze der additiven Zahlentheorie ’, Math. Ann. 57 (1934), 668–678.
Zhi-Wei Sun, On integers not of the form $\pm p^a\pm q^b$, Proc. Amer. Math. Soc. 128 (2000), no. 4, 997–1002. MR 1695111, DOI 10.1090/S0002-9939-99-05502-1