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It is shown how polynomial time prime tests, which are both fast and deterministic, can be developed for many numbers of the form      A r n −  1 ( r = 5 , 7 ; A &gt; r n ) A{r^n} - 1\;(r = 5,7;A &gt; {r^n})      . These tests, like the Lucas-Lehmer test for the primality of the Mersenne numbers, are derived by using the properties of the Lucas functions. We exemplify these ideas by using numbers of the form      2 ⋅  10 n −  1 2 \cdot {10^n} - 1      .

Effective primality tests for some integers of the forms 𝐴5ⁿ-1 and 𝐴7ⁿ-1