哥德巴赫与杜布纳猜想之间的一个中间猜想：每个偶数都是一个素数与一个孪生素数对中一项之和
An intermediate conjecture between Goldbach and Dubner: every even number is the sum of a prime and a twin prime
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中文总结 AI 辅助
该研究提出介于哥德巴赫与杜布纳猜想间的中间猜想，用罗曼诺夫方法结合塞尔伯格筛法给出条件性结果，验证了10^14以内偶数均满足该猜想。
中文摘要 AI 辅助
我们研究这样一个命题：每个满足 $n \ge 6$ 的偶数都是一个素数与孪生素数对中的一项之和。该命题介于哥德巴赫猜想与杜布纳猜想之间，且蕴含这两个猜想。我们的主要结果是条件性的：若对于所有足够大的 $z$，不假设孪生素数的分布，仅要求不超过 $z$ 的孪生素数个数至少为 $c\\,z/\log^2 z$（这是哈代与李特尔伍德预测量级的下界），则存在正比例的偶数可表示为该形式，其密度至少为 $c$ 的绝对倍数。证明采用罗曼诺夫方法与塞尔伯格筛法，未损失任何 $\log\log$ 因子。该方向上不存在无条件结果，因为正密度集合上的可表示性已蕴含孪生素数猜想；我们进一步证明，最小孪生素数加项的多对数界将迫使孪生素数个数的幂次型下界。我们对所有不超过 $10^{14}$ 的偶数穷尽验证了该命题，其中最小孪生素数加项从未超过23029。
英文摘要
We study the statement that every even number $n \ge 6$ is the sum of a prime and a member of a twin prime pair. It sits between the conjectures of Goldbach and Dubner and implies both the Goldbach and the twin prime conjectures. Our main result is conditional: if the number of twin primes up to $z$ is at least $c\,z/\log^2 z$ for all large $z$, a lower bound of the order predicted by Hardy and Littlewood, with nothing assumed about their distribution, then a positive proportion of the even numbers are so representable, with density at least an absolute multiple of $c$. The proof is Romanov's method with a Selberg sieve, and loses no factor of $\log\log$. Nothing in this direction can be unconditional, since representability on a set of positive density already implies the twin prime conjecture; we show further that a polylogarithmic bound on the least twin summand would force a power-type lower bound on the number of twin primes. We verify the statement exhaustively for all even numbers up to $10^{14}$, where the least twin summand never exceeds 23,029.
