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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, aresult known as Lucas's converse of Fermat's little theorem. This result wasconsiderably improved by Henry Pocklington in 1914 when he showed that it's notnecessary 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 integersof the form $ap^{k}+1$ where $p$ is prime, $a<p$, $k\ge 1$. An extension ofLucas's converse of Fermat's little theorem is given. We also prove a newgeneral-purpose primality test that requires that only a single odd primedivisor of $n-1$ be found for the test to be implemented. Contrary to thewell-known result: There are infinitely many Fermat pseudoprimes to any base$b$; In this paper we prove the finitude of Fermat pseudoprimes in some formsof integers.

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