Abstract
The Lucas primality test is a procedure that verifies an odd integer \(n\) for primality using the condition \(F_{n-e(n)}\equiv\ 0\;(\bmod\ n)\), where \(\{F_{n}\}_{n}\) is the Fibonacci series, \(e(n)=\binom{n}{5}\) is the Legendre symbol. In this paper, we study a modified Lucas test called the Lucas–Miller–Rabin primality test (briefly, LMR test) and demonstrate that the LMR test is much more efficient than the original Lucas test. We describe necessary and sufficient conditions for composite integers to pass the LMR test. Composite integers passing it are called LMR-pseudoprimes. We show that the existence of LMR-pseudoprimes divisible by squares depends on the validity of the Wall–Sun–Sun Hypothesis, one of the long-standing problems in the Number Theory.
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Funding
This paper has been supported by the Kazan Federal University Strategic Academic Leadership Program (‘‘PRIORITY-2030’’), Strategic Project no. 4.
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(Submitted by A. M. Elizarov)
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Ishmukhametov, S., Antonov, N. & Mubarakov, B. On a Modification of The Lucas Primality Test. Lobachevskii J Math 44, 2700–2706 (2023). https://doi.org/10.1134/S1995080223070193
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DOI: https://doi.org/10.1134/S1995080223070193