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.

中文翻译：

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素数之间最可能的算术级数差异

令\（d ^ * _ k（x）\）是素数\（\ le x \）之间长度\（k + 1 \）的算术级数最可能的常见差异。根据Hardy–Littlewood猜想的真相，我们得出\（\ lim \ limits _ {x \ rightarrow + \ infty} d ^ * _ k（x）= + \ infty \）均匀地分布在k中，并且每个素数都将其除以足够大的最可能的共同差异。