On integers of the form p+2^k

Romanov type problems

“…This implies that the proportion of such n with a representation of the form is zero. We have choices for the residue class a and half of the integers in these residue classes are odd which yields a density ofWe note that a more refined version of the above argument was used by Habsieger and Roblot [11, Sect. 3] to prove an upper bound on the proportion of odd integers not of the form .…”

“…We note that a more refined version of the above argument was used by Habsieger and Roblot [11,Sect. 3] to prove an upper bound on the proportion of odd integers not of the form p + 2 k .…”

“…In both cases this yields a contradiction to n ≡ 3 mod 6. For k ≥ 2 and q odd, we have that 2 2 k ≡ {16, 24, 25} mod 29 and 2 q ≡ {2, 3,8,10,11,12,14,15,17,18,19,21, 26, 27} mod 29. For k ≥ 2 and q odd, it is thus true that 2 2 k + 2 q ≡ 0 mod 29 and thus n = 2 + 2 2 k + 2 q ≡ 2 mod 29 yields a contradiction in this case.…”

ALMOST-PRIMES REPRESENTED BY p + am

“…For example, Chen and Sun [1] proved that c > 0.0868, this result is improved by Habsieger and Roblot [6] to 0.0933 and by Pintz [7] to 0.09368.…”

The Romanoff theorem revisited

“…A classical result of Romanoff [6] asserts that the sumset 2 N + P = {2 n + p : n ∈ N, p ∈ P} has a positive lower density, i.e., there exists a positive constant C R such that (2 N + P)(x) ≥ C R x for sufficiently large x. Recently, the lower bound of C R has been calculated in [2,3,5]. Now let P 2 = {q : q is a prime or the product of two primes}.…”