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A primality test for $$4Kp^n-1$$ numbers. (English) Zbl 1461.11018
Classical Lucasian type primality tests provide primality criteria for an integer $$N$$ written in a specific form and based on the use of a recursive sequence. The initial value, $$S_0$$, of the recursion usually depends on the specific parameter values used to represent $$N$$. Motivated by this dependence of $$S_0$$ on the specific $$N$$, the authors provide an alternative in which the necessity is dropped in order to obtain a laxer sufficient primality condition independent of any parameters of $$N$$ in a given parametrised set of integers.
The authors present in Corollary 1 a Lucasian type primality test for numbers written in the form $$N=4 K p^n - 1$$, in which $$p$$ is an odd prime, $$n\geq 1$$ and $$K$$ an odd integer with $$4K \leq p^n$$. Moreover, in Corollary 2 and setting $$p=2$$, a Gaussian analogue to the classical Lucas-Lehmer-Riesel test is presented. In Section 4 the computational complexity of their proposed algorithm is discussed. Finally, in Section 5 the algorithm is compared to other methods and the probability with which a prime is correctly certified by the algorithm is discussed as well.
It is noted that similar results have been obtained by different authors [E. L. Roettger et al., Des. Codes Cryptography 77, No. 2-3, 515–539 (2015; Zbl 1364.11161)], but that the presentation of the results is novel and original.
MSC:
 11A51 Factorization; primality 11Y11 Primality 11Y40 Algebraic number theory computations
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