Primes as sums of Fibonacci numbers

Abstract

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 expansion of a positive integer n -we write n as the sum of non-consecutive Fibonacci numbers. More precisely, we are concerned with the Zeckendorf sum-of-digits function z, which returns the minimal number of Fibonacci numbers needed to write a positive integer as their sum. The proof of such a prime number theorem for z, 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 driving forces of this development were the Gelfond problems (1968/1969), more specifically the behavior of the sum-of-digits function in base q along the sequence of primes and along integervalued polynomials, and the Sarnak conjecture. Mauduit and Rivat resolved the question on the sum of digits of prime numbers (2010) and the sum of digits of squares (2009). 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.For the (partial) solution of the Gelfond problems, Mauduit and Rivat have developed a powerful method that is based on cutting-off-digitstechniques, on sophisticated estimates for Fourier terms, and on estimates for exponential sums. These techniques -together with a new decomposition of finite automata -were also the basis for the second author's result on automatic sequences.The main new technical tools that are used in order to obtain corresponding results for Fibonacci numbers are Gowers norms, and a significant extension of Mauduit and Rivat's method. Gowers norms are a proper generalization of the above-mentioned Fourier terms; employing them in our proof allows us to prove that exp(2πiαz(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.

Möbius orthogonality of Thue-Morse sequence along Piatetski-Shapiro numbers

“…Sketch of the proof. The proof of Theorem 1.1 is based on the techniques developed recently by Spiegelhofer for the level of distribution of Beatty subsequences t(⌊αn + β⌋) [39] and by Drmota-Müllner-Spiegelhofer for primes along the sums of Fibonacci numbers [17].…”

“…this is apparently beyond the limitation of the approaches from [39] and [17]. Möbius orthogonality is an easier problem as there are more tools for decomposing the original sum with µ(n) compared to the case of Λ(n).…”

“…The precise version of this property is given by Lemma 4.3 (carry propagation lemma). It was introduced in the works of Maudit and Rivat [34,35], and later used in [17,37,39]. The important advantage of the truncated function is its periodicity.…”

“…It allows one to explore the distribution of s λ (n) by looking at the fractional part of n/2 λ , which can be handled by a Fourier analysis. A somewhat similar idea is used in [17] for the sums of Fibonacci numbers where one needs to study the fractional part of γn, where γ is the golden ratio. Going back to the Thue-Morse case, the distribution of {n/2 λ } is easier to handle when λ is small, as the corresponding subintervals of [0, 1) are large.…”

“…However, the error terms produced by carry propagation lemma force a large choice of λ. We truncate the funtion s λ a few more times using the techniques for "cutting away" of the digits from [17,39]. There are limitations on how many digits one can cut at each step, and this is the reason why we apply the generalized van der Corput inequality several times.…”