On the sum of a prime and two Fibonacci numbers

2025/06/04 by Ji-Zhen Xu, Xu, Ji-Zhen, Yong-Gao Chen +1
Computer Science · Mathematics · Physics and Astronomy · #11A41 #11B13 #11B39 #11P32 #Advanced Mathematical Theories and Applications #Analytic Number Theory Research #FOS: Mathematics #Number Theory (math.NT) #semigroups and automata theory

paper · pdf · doi:10.48550/arxiv.2506.03631

openalex publication_date 2025/06/04 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let \f_n\ be the Fibonacci sequence. For any positive integer n, let r(n) be the number of solutions of n=p+f_k1² +f_k₂², where p is a prime and k₁, k₂ are nonnegative integers with k₁≤ k₂. In this paper, it is proved that \ n : r(n)=0\ contains an infinite arithmetic progression, and both sets \ n : r(n)=1\ and \ n : r(n)≥ 2\ have positive asymptotic densities.

Citations

• Quantitative results of the Romanov type representation functions
• On the integers not of the form p+2^a+2^b
• Approximate formulas for some functions of prime numbers

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• On the Diophantine equation ∑ _k=1⁵F_nk=2^a
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