Primes as sums of Fibonacci numbers

Michael Drmota, Clemens Müllner, Lukas Spiegelhofer

Published 2021-09-09Version 1

The purpose of this paper is to discuss the relationship between prime numbers and sums of Fibonacci numbers. One of our main results says that for every sufficiently large integer $k$ there exists a prime number that can be represented as the sum of k different and non-consecutive Fibonacci numbers. This property is closely related to, and based on, a prime number theorem for certain morphic sequences. In our case, these morphic sequences are based on the Zeckendorf sum-of-digits function $z$, which returns the number of summands in the representation of a nonnegative integer as sum of non-consecutive Fibonacci numbers. The proof of such a prime number theorem, combined with a corresponding local result, constitutes the central contribution of this paper, from which the result stated in the beginning follows. These kinds of problems have been discussed intensively in the context of the base-$q$ expansion of integers. The Gelfond problems (1968/1969), and the Sarnak conjecture, were the driving forces of this development. Mauduit and Rivat resolved the question on the sum of digits of prime numbers (2010) and the sum of digits of squares (2009), thus leaving open only part of the third Gelfond problem. Later the second author (2017) proved Sarnak's conjecture for the class of automatic sequences, which are based on the $q$-ary expansion of integers, and which generalize the sum-of-digits function in base $q$ considerably. In the present paper, we use Gowers norms, and extend Mauduit and Rivat's method significantly. Employing Gowers norms in our proof allows us to prove that $\exp(2\pi i \alpha z(n))$ has level of distribution 1. This latter result forms an essential part of our treatment of the occurring sums of type I and II. An analogous result for the Thue--Morse sequence has been obtained by the third author (2020).