  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.
