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 $\widehat O(\log^2N)$. 
@article{DM_2008_20_2_a1,
     author = {E. V. Sadovnik},
     title = {Testing numbers of the form $N=2kp_1^{m_1}p_2^{m_2}\cdots p_n^{m_n}-1$ for primality},
     journal = {Diskretnaya Matematika},
     pages = {15--24},
     publisher = {mathdoc},
     volume = {20},
     number = {2},
     year = {2008},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/DM_2008_20_2_a1/}
}
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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/