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We propose an algorithm determining the primality of numbers $M=Ap^n+w_n$where $w_n^{p-1}\equiv1\pmod{p^n}$ and $A<p^n$ and give example when $p=7$. $p$th reciprocity law is involved. The algorithm runs in polynomial time in$\log_2(M)$ for fixed $p$ and $A$.

Primality Test for Numbers of the Form Ap^n+w