On integers of the form $p+F_{2^k}+F_q$

Yang Gao

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arXiv · 2606.30188

On integers of the form $p+F_{2^k}+F_q$

Yang Gao

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Abstract

In 1934, Romanoff proved that the set of positive integers representable as the sum of a prime and a power of two has positive lower density. Erd\H{o}s later constructed an infinite arithmetic progression of odd integers none of which admits such a representation. Let $F_n$ be the Fibonacci sequence. In this paper, we prove that the set of integers of the form $p+F_{2^k}+F_q$, where $p,q$ are primes and $k\ge0$, has positive lower asymptotic density. The same holds for the set of integers not of this form.

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BibTeX RIS

Yang Gao. 2026-06-29. On integers of the form $p+F_{2^k}+F_q$. https://arxiv.org/abs/2606.30188

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