发表机构
HSE University（高等经济大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究改进素数加2的幂形式整数的长间隙下界，通过截断除数积和推广平移引理，将间隙下界提升至$\log\log X$，并推广至$p+a^n$情形。
AI 中文摘要
关于整数表示为素数加2的幂之和的问题，可追溯至1752年欧拉与哥德巴赫之间的通信以及德·波利尼亚克1849年的工作。Romanoff证明了具有这种表示的整数集合具有正的下密度。沿着Kalmynin和Konyagin研究的互补方向，我们考虑不包含Romanoff数的长区间。我们利用除数和的截断乘积来推导出在$[1,X]$内不包含形如$p+2^n$（其中$p$为素数且$n\ge1$）的整数的最长连续区间的下界$G_{\mathcal{R}}(X)\gg\log\log X$。这改进了Kalmynin和Konyagin的界$G_{\mathcal{R}}(X)\gg\log\log X/\log\log\log X$。关键步骤是将一个平移引理推广到至多$X$的绝对值的$O(\log X)$个整数平移的集合：以可忽略的代价移除整除平移差的辅助素数。在初步筛分之后，这使得所有剩余的平移能够被同时覆盖。该论证是无条件的，并且也适用于每个固定整数$a\ge2$的$p+a^n$形式。
英文摘要
Questions concerning representations of integers as the sum of a prime and a power of two go back to the correspondence between Euler and Goldbach in 1752 and to de Polignac's work of 1849. Romanoff proved that the set of integers admitting such a representation has positive lower density. Following the complementary direction studied by Kalmynin and Konyagin, we consider long intervals containing no Romanoff numbers. We use truncated products of divisor sums to derive the bound $G_{\mathcal{R}}(X)\gg\log\log X$ for the longest block in $[1,X]$ containing no integer of the form $p+2^n$, with $p$ prime and $n\ge1$. This improves the bound $G_{\mathcal{R}}(X)\gg\log\log X/\log\log\log X$ of Kalmynin and Konyagin. The key step extends a translate lemma to sets of $O(\log X)$ integer shifts of absolute value at most $X$: auxiliary primes dividing differences of shifts are removed at negligible cost. After preliminary sieving, this allows all remaining shifts to be covered simultaneously. The argument is unconditional and also applies to $p+a^n$ for every fixed integer $a\ge2$.
