Mixed sums of primes and other terms

TL;DR

This paper investigates properties of mixed sums involving primes and linear recurrences, proving specific non-equality results and conjecturing representations of integers as sums of primes and Fibonacci numbers.

Contribution

It introduces new results on sums of primes and linear recurrences, including non-equality theorems and a conjecture on representing integers as sums involving primes and Fibonacci numbers.

Findings

Proves that certain prime-related expressions are not equal to specific powers.

Shows all sums of the form u_m + a u_n are distinct for a linear recurrence.

Conjectures that all integers greater than 4 can be expressed as a sum of a prime and Fibonacci numbers.

Abstract

In this paper we study mixed sums of primes and linear recurrences. We show that if m=2(mod 4) and m+1 is a prime then $(m^{2^n-1}-1)/(m-1)\not=m^n+p^a$ for any n=3,4,... and prime power p^a. We also prove that if a>1 is an integer, u_0=0, u_1=1 and u_{i+1}=au_i+u_{i-1} for i=1,2,3,..., then all the sums u_m+au_n (m,n=1,2,3,...) are distinct. One of our conjectures states that any integer n>4 can be written as the sum of an odd prime, an odd Fibonacci number and a positive Fibonacci number.