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- W3030284145 abstract "In 1876, Edouard Lucas showed that if an integer $b$ exists such that $b^{n-1}equiv 1, (mathrm{mod}, n)$ and $b^{(n-1)/q}mathbf{notequiv }1, (mathrm{mod}, n)$ for all prime divisors $q$ 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 $n-1$ to determine the primality of $n$. In this paper we optimize Pocklington's primality test for integers of the form $ap^{k}+1$ where $p$ is prime, $a<4(p+1)$, $kge 1$. Precisely, this paper shows that if an integer $b$ exists such that $b^{n-1}equiv 1, (mathrm{mod}, n)$ and $n nmid b^{(n-1)/p}-1$, then $n$ is prime as opposed to Pocklington's primality test that imposes the more stringent hypothesis that $n$ and $b^{(n-1)/p}-1$ be relatively prime. Based on substantial experimental data, the reader is invited to extend this result for all positive integers $n=am+1$, $a<4(p+1)$ where $p$ is the least prime divisor of $m$." @default.
- W3030284145 created "2020-06-05" @default.
- W3030284145 creator A5035793856 @default.
- W3030284145 date "2020-05-04" @default.
- W3030284145 modified "2023-09-24" @default.
- W3030284145 title "A note on primality of $ap^{k}+1$ numbers" @default.
- W3030284145 hasPublicationYear "2020" @default.
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