Geodesic
On a conbined primality test
Izvestiâ vysših učebnyh zavedenij. Matematika, no. 12 (2022), pp. 123-129

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In this paper we consider a hybrid primality test consisting of checking the relation $2^{n-1}\equiv 1 (\bmod\ n)$ and the Lucas primality test. Let call this procedure as $\mathrm{L}2$-test. Composite integers passing $\mathrm{L}2$-test are called $\mathrm{L}2$-pseudoprime. In this paper we develop an effective algorithm for searching $\mathrm{L}2$-pseudoprimes of form $n\equiv\pm 2(\bmod 5)$. Using it we prove that there are no $\mathrm{L}2$-pseudoprimes of the mentioned form below $B=10^{23}$ (it is the currently reached boarder and it continues to increase). Thus, $\mathrm{L}2$-test is a deterministic test at the current interval up to $B=10^{23}$ allowing the researchers to check an odd $n\equiv\pm 2(\bmod 5)$ for primality using a polynomial two-round procedure of rate $O(\ln^3 n)$.
Keywords: Lucas primality test, the Fermat test, probabilistic primality test, deterministic primality test. 
@article{IVM_2022_12_a10,
     author = {Sh. T. Ishmukhametov and N. A. Antonov and B. G. Mubarakov and G. G. Rubtsova},
     title = {On a conbined primality test},
     journal = {Izvesti\^a vys\v{s}ih u\v{c}ebnyh zavedenij. Matematika},
     pages = {123--129},
     publisher = {mathdoc},
     number = {12},
     year = {2022},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/IVM_2022_12_a10/}
}
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Sh. T. Ishmukhametov; N. A. Antonov; B. G. Mubarakov; G. G. Rubtsova. On a conbined primality test. Izvestiâ vysših učebnyh zavedenij. Matematika, no. 12 (2022), pp. 123-129. http://geodesic.mathdoc.fr/item/IVM_2022_12_a10/