On Romanoff's theorem

In 1934 N. P. Romanoff proved the following theorem. Let $a$ be a positive integer with $a>1$. Then there is a number $c(a)>0$, depending only on $a$, such that
$$ \#\{ 1\leq n\leq x:\ \hbox{there is a prime p and a non-negative integer j such that} p+a^j=n \}\geq c(a)x $$
for $x\geq 4$. We will discuss some results related to Romanoff's theorem.
Conference ID: 942 0186 5629 Password is a six-digit number, the first three digits of which form the number p + 44, and the last three digits are the number q + 63, where p, q is the largest pair of twin primes less than 1000