Primes as Sums of Fibonacci Numbers

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 k there exists a prime number that can be represented as the sum of k 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 n — we write n n as the sum of non-consecutive Fibonacci numbers. More precisely, we are concerned with the Zeckendorf sum-of-digits function z 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 z, combined with a corresponding local result, constitutes the central contribution of this paper, from which the result stated in the beginning follows. Problems of this type have been discussed intensively in the context of the base- q 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 q along the sequence of primes and along integer-valued 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 q -ary expansion of integers, and which generalize the sum-of-digits function in base q q considerably. For the (partial) solution of the Gelfond problems (1967/1968), Mauduit and Rivat have developed a powerful method that is based on techniques for “cutting off digits”, 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. In order to obtain corresponding results for Fibonacci numbers, we have to extend Mauduit and Rivat’s method considerably. In fact, we are departing significantly from this method, proving the statement that exp ⁡ (2 π i ϑ z (n) ) (2 i 0. 5pt z (n) ) has level of distribution 1 1. This latter result forms an essential part of our treatment of the occurring sums of type I I and I I II and uses Gowers norms related to the Zeckendorf sum-of-digits function as a central technical tool. Gowers norms are a higher order generalization of the above-mentioned Fourier terms, and their appearance in our method is intimately tied to the iterated application of a new generalization of van der Corput’s inequality.