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我们提出了一种新的分析方法,基于质数的$6x \pm 1$表示来研究孪生质数猜想。 通过定义所谓的孪生质数生成器$x \in \N$,其中$6x - 1$和$6x + 1$均为质数,我们将该猜想重新表述为这类$x$的存在性问题。 利用小质数乘积的容许剩余类和调整后的塞尔伯格筛法,我们将自然数划分为结构化的区间$\mc{A}_n$,其中$6x \pm 1$的最大可能质因数是固定的。 在每个$\mc{A}_n$中,我们应用筛法来估计逃过所有局部障碍的生成器候选数量。 我们推导出筛法主项和误差项的显式渐近界,并证明主项占主导地位。 这导致了这样的结果,即存在无限多个区间$\mc{A}_n$包含超过一个孪生质数生成器。 这表明孪生质数是无限的。
We present a novel analytic approach to the Twin Prime Conjecture based on the $6x \pm 1$ representation of primes. By defining so-called twin prime generators $x \in \N$, for which both $6x - 1$ and $6x + 1$ are prime, we reformulate the conjecture into the existence problem of such $x$. Using admissible residue classes modulo products of small primes and an adapted Selberg sieve, we partition the natural numbers into structured intervals $\mc{A}_n$, where the maximal possible prime divisor of $6x \pm 1$ is fixed. Within each $\mc{A}_n$, we apply the sieve to estimate the number of generator candidates that escape all local obstructions. We derive explicit asymptotic bounds for the main and error terms of the sieve and prove that the main term dominates. This leads to the result, that infinitely many intervals $\mc{A}_n$ contain more than one twin prime generator. This implies the infinitude of the twin primes.
使用快速傅里叶变换,我们提出了一种可以直接使用的对称循环实线性方程组的解决方案,特别适用于涉及更广泛理论分析的情况。
Employing the Fast Fourier Transform we propose a ready-to-use solution to symmetric circulant real linear systems of equations, particularly useful when a broader theoretical analysis is involved.