德-波利尼亚克关于 p + 2k 形式数的猜想的一些计算结果

Pub Date : 2024-08-20 DOI:10.1016/j.jnt.2024.07.004
Yuda Chen, Xiangjun Dai, Huixi Li
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引用次数: 0

摘要

设 U 是不能写成 p+2k 形式的正奇数的集合。最近,Chen 通过分析 b 的可能素除数,证明了如果算术级数 a(modb) 在 U 中,则 b≥11184810 且 ω(b)≥7,当且仅当 b=11184810 时,ω(b)=7,其中 ω(n) 是 n 的不同素除数的个数。本文采用计算方法证明 b≥11184810,并提供了 a(mod11184810) 在 U 中时 a 的所有可能值。此外,我们明确地构造了 U 中 ω(b)=8, 9, 10 或 11 的非微不足道的算术级数 a(modb),并提供了 U 中潜在的非微不足道的算术级数 a(modb),使得任何固定的 s≥12 时,ω(b)=s。此外,我们通过增强算法和使用 GPU 计算,将哈布西格和罗布洛 2006 年对 p+2k 形式数的估计上限提高到了 0.490341088858244。
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Some computational results on a conjecture of de Polignac about numbers of the form p + 2k

Let U be the set of positive odd numbers that can not be written in the form p+2k. Recently, by analyzing possible prime divisors of b, Chen proved b11184810 and ω(b)7 if an arithmetic progression a(modb) is in U, with ω(b)=7 if and only if b=11184810, where ω(n) is the number of distinct prime divisors of n. In this paper, we take a computational approach to prove b11184810 and provide all possible values of a if a(mod11184810) is in U. Moreover, we explicitly construct nontrivial arithmetic progressions a(modb) in U with ω(b)=8, 9, 10, or 11, and provide potential nontrivial arithmetic progressions a(modb) in U such that ω(b)=s for any fixed s12. Furthermore, we improve the upper bound estimate of numbers of the form p+2k by Habsieger and Roblot in 2006 to 0.490341088858244 by enhancing their algorithm and employing GPU computation.

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