On a Weakened Form of Polignac Conjecture

Polignac [1] conjectured that for every even natural number $2k (k\geq1)$, there exist infinitely many consecutive primes $p_n$ and $p_{n+1}$ such that $p_{n+1}-p_n=2k$. A weakened form of this conjecture states that for every $k\geq1$, there exist infinitely many primes $p$ and $q$ such that $p-q=2k$. Clearly, the weakened form of Polignac's conjecture implies that there exists an infinite sequence of positive integers $x_1,x_2,...,x_m,...$ such that $x_1 (2k+x_1),x_2 (2k+x_2),...,x_m (2k+x_m)...$ are pairwise relatively prime. In this note, we obtain a slightly stronger result than this necessary condition. This enables us to find a common property on some special kinds of number-theoretic functions (such as $2^x -1$) which likely represent infinitely many primes by rich literatures and a lot of research reports. However, the function $2^{2^x}+1$ does not have this property. Does it imply that the number of Fermat primes is finite? Hardy and Wright [7] conjectured that the number of Fermat primes is finite. Nevertheless, they did not give any reasons and explanations. By factoring Fermat number, many people believe that the conjecture in [7] holds. Does our work explain this phenomenon? We will consider further this problem in another paper. Based on our work, one could give a new sufficient condition that there are an infinite number of twin primes (Sophie-Germain primes or Mersenne primes).