arXiv 2005.023272020-05math.NT

$ap^{k}+1$形式数的素性

Primality of numbers of the form $ap^{k}+1$

Ariko Stephen Philemon

摘要：

1876年，爱德华·卢卡斯证明了如果存在一个整数$b$，使得对于$n-1$的所有素因子$p$，$b^{n-1} \equiv 1 (\mathrm{mod} \ n)$和$b^{(n-1)/p} \not\equiv 1( \mathrm{mod} \ n)$都是，那么$n$是素的，这个结果被称为卢卡斯对费马小定理的逆。这个结果在1914年被Henry Pocklington大大改进，他证明了不需要知道$n-1$的所有素因子就可以确定$n$是否是素数。在本文中，我们优化Pocklington的素数测试的形式$ap^{k}+1$，其中$p$是素数，$a<p$，$k\ge 1$。本文给出了卢卡斯关于费马小定理的逆定理的一个推广。我们还证明了一个新的通用素性测试，要求只有一个单一的奇素数因子的$n-1$被发现的测试实施。与著名的结论相反：任何基$b$都有无穷多个Fermat伪素数;本文证明了整数形式的Fermat伪素数的有限性。

英文摘要：

In 1876, Edouard Lucas showed that if an integer $b$ exists such that $b^{n-1} \equiv 1 (\mathrm{mod} \ n)$ and $b^{(n-1)/p} \not\equiv 1( \mathrm{mod} \ n)$ for all prime divisors $p$ of $n-1$ , then $n$ is prime, a result known as Lucas's converse of Fermat's little theorem. This result was considerably improved by Henry Pocklington in 1914 when he showed that it's not necessary to know all the prime factors of $n-1$ in order to determine if $n$ is prime. In this paper we optimize Pocklington's primality test for integers of the form $ap^{k}+1$ where $p$ is prime, $a<p$, $k\ge 1$. An extension of Lucas's converse of Fermat's little theorem is given. We also prove a new general-purpose primality test that requires that only a single odd prime divisor of $n-1$ be found for the test to be implemented. Contrary to the well-known result: There are infinitely many Fermat pseudoprimes to any base $b$; In this paper we prove the finitude of Fermat pseudoprimes in some forms of integers.

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