The Most Likely Common Difference of Arithmetic Progressions Among Primes

Abstract

Let $d^*_k(x)$ be the most likely common differences of arithmetic progressions of length $k+1$ among primes $\le x$. Based on the truth of Hardy–Littlewood Conjecture, we obtain that $\lim \limits _{x\rightarrow +\infty }d^*_k(x)=+\infty $ uniformly in k, and every prime divides all sufficiently large most likely common differences.

Keywords

Common difference / Arithmetic progression / Hardy–Littlewood Conjecture / Differences among primes / Singular series

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Funding

National Natural Science Foundation of China(11871187)