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Transcendental series of reciprocals of Fibonacci and Lucas numbers

Khoa Dang Nguyen

Vol. 16 (2022), No. 7, 1627–1654
Abstract

Let F1 = 1,F2 = 1, be the Fibonacci sequence. Motivated by the identity k=01F2k = 7 52, Erdös and Graham asked whether k=11Fnk is irrational for any sequence of positive integers n1,n2, with nk+1nk c > 1. We resolve the transcendence counterpart of their question; as a special case of our main theorem, we have that k=11Fnk is transcendental when nk+1nk c > 2. The bound c > 2 is best possible thanks to the identity at the beginning. This paper provides a new way to apply the subspace theorem to obtain transcendence results and extends previous nontrivial results obtainable by only Mahler’s method for special sequences of the form nk = dk + r.

Keywords
algebraic numbers, transcendental numbers, subspace theorem
Mathematical Subject Classification
Primary: 11J87
Secondary: 11B39
Milestones
Received: 27 November 2020
Revised: 20 September 2021
Accepted: 23 October 2021
Published: 16 October 2022
Authors
Khoa Dang Nguyen
Department of Mathematics and Statistics
The University of Calgary
Calgary AB
Canada