Geodesic
Testing numbers of the form $N=2kp^m-1$ for primality
Diskretnaya Matematika, Tome 18 (2006) no. 1, pp. 146-155.

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We suggest an algorithm to test numbers of the form $N=2kp^m-1$ for primality, where $2k$, $k$ is an odd positive integer, $2k$, $p$ is a prime number, and $p=3\pmod 4$. The algorithm makes use of the Lucas functions. First we present an algorithm to test numbers of the form $N=2k3^m-1$. Then the same technique is used in the more general case where $N=2kp^m-1$. The algorithms suggested here are of complexity $O((\log N)^2 \log\log N \log\log\log N)$. 
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E. V. Sadovnik. Testing numbers of the form $N=2kp^m-1$ for primality. Diskretnaya Matematika, Tome 18 (2006) no. 1, pp. 146-155. http://geodesic.mathdoc.fr/item/DM_2006_18_1_a10/

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