Geodesic
Testing numbers of the form $N=2kp_1^{m_1}p_2^{m_2}\cdots p_n^{m_n}-1$ for primality
Diskretnaya Matematika, Tome 20 (2008) no. 2, pp. 15-24.

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We suggest an algorithm to test numbers of the form $N=2kp_1^{m_1}p_2^{m_2}\cdots p_n^{m_n}-1$ for primality, where $2k$, $k$ is an odd positive integer, $\pi$ is a prime number, $i=1,\dots,n$, and $p_1p_2\cdots p_n=3\pmod4$. The algorithm makes use of the Lucas functions. The algorithm suggested is of complexity $\widehat O(\log^2N)$. 
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E. V. Sadovnik. Testing numbers of the form $N=2kp_1^{m_1}p_2^{m_2}\cdots p_n^{m_n}-1$ for primality. Diskretnaya Matematika, Tome 20 (2008) no. 2, pp. 15-24. http://geodesic.mathdoc.fr/item/DM_2008_20_2_a1/

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