On the sum of two primes and k powers of two A. LANGUASCO, J. PINTZ, A. ZACCAGNINI Abstract Let X be a large integer. We prove that, for any fixed positive integer k,a suitable asymptotic formula for the number of representations of an even inte- ger N ∈ [1,X ] as the sum of two primes and k powers of 2 holds with at most O k ( X 3/5 (log X ) 10 ) exceptions. 2000 Mathematics Subject Classification: 11P32 (primary), 11P55 (secondary). This is the last preprint. The final paper will appear in the Bull. London Math. Soc., 39:771780, 2007. 1 Introduction Throughout the paper, k denotes a fixed positive integer. Constants implied by the O(·) and ≪ notations will silently depend on k. We will use X to denote a large parameter, L for log 2 X (the base 2 logarithm of X ) and J (X ) for the interval [2X/3,X ]. Our main concern is the representation of an even integer N ∈J (X ) as a sum of two primes and k powers of two: we let r ′′ k (N )= |{(p 1 ,p 2 ,ν 1 ,...,ν k ) ∈ P 2 × [1,L] k : N = p 1 + p 2 +2 ν 1 + ··· +2 ν k }| denote the relevant counting function, where P is the set of all prime numbers. The problem of estimating r ′′ k (N ) is sometimes considered as an approximation to Goldbach’s Conjecture. We recall that the first result on this topic was proved by Linnik [10, 11]: there exists a constant k> 0 such that r ′′ k (N ) > 0 for every large even integer N . Then the next step was made by Gallagher [2]: for any k ≥ 2 there exists N k such that, for every even integer N ≥ N k , one has r ′′ k (N )=2 N log k 2 N log 2 N  1+ O  log 2 k k  . We remark that Gallagher’s result is not an asymptotic formula because k is a fixed integer and hence the “error term” in the previous formula has the same order as the “main term”. Unfortunately there was no specified numerical value for k in these results. In the late nineties, in a series of paper by Liu, Liu & Wang [12, 13, 14], Wang [23] and Li [8, 9], it was proved that, if k ≥ k 0 , then r ′′ k (N ) > 0 for every sufficiently large N, (1) where k 0 is an explicitly given constant. Such estimates for k 0 were determined both conditionally (i.e. assuming GRH) and unconditionally. In the unconditional case the 1