Mixed sums of primes and other terms

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 2n-1-1)/(m - 1) ≠ mn + pa for any n = 3, 4, ⋯ and prime power pa. We also prove that if a > 1 is an integer, u 0 = 0, u 1 = 1, and ui+1 = au i + ui-1 for i = 1, 2, 3, ⋯, then all the sums um + aun (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 and two positive Fibonacci numbers. © 2010 Springer Science+Business Media, LLC.