Lucasian criteria for the primality of

Author:
Hans Riesel

Journal:
Math. Comp. **23** (1969), 869-875

MSC:
Primary 10.08

MathSciNet review:
0262163

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Abstract: Let with given. If is a necessary and sufficient criterion that be prime, this is called a Lucasian criterion for the primality of . Many such criteria are known, but the case has not been treated in full generality earlier. A theorem is proved that (by aid of computer) enables the effective determination of suitable numbers for any given , if . The method is used on all in the domain . The Lucasian criteria thus constructed are applied, and all primes in the domain are tabulated.

**[1]**E. L. Ince,*Cycles of Reduced Ideals in Quadratic Fields*, British Association for the Advancement of Science, London, 1934.**[2]**D. H. Lehmer, ``An extended theory of Lucas' functions,''*Ann. of Math.*, v. 31, 1930, pp. 419-448.**[3]**Edouard Lucas,*Theorie des Fonctions Numeriques Simplement Periodiques. [Continued]*, Amer. J. Math.**1**(1878), no. 3, 197–240 (French). MR**1505164**, 10.2307/2369311**[4]**Hans Riesel,*A note on the prime numbers of the forms 𝑁=(6𝑎+1)2²ⁿ⁻¹-1 and 𝑀=(6𝑎-1)2²ⁿ-1*, Ark. Mat.**3**(1956), 245–253. MR**0076793****[5]**H. C. Williams and C. R. Zarnke,*A report on prime numbers of the forms 𝑀=(6𝑎+1)2^{2𝑚-1}-1 and 𝑀′=(6𝑎-1)2^{2𝑚}-1*, Math. Comp.**22**(1968), 420–422. MR**0227095**, 10.1090/S0025-5718-1968-0227095-2**[6]**Raphael M. Robinson,*A report on primes of the form 𝑘⋅2ⁿ+1 and on factors of Fermat numbers*, Proc. Amer. Math. Soc.**9**(1958), 673–681. MR**0096614**, 10.1090/S0002-9939-1958-0096614-7

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DOI:
http://dx.doi.org/10.1090/S0025-5718-1969-0262163-1

Article copyright:
© Copyright 1969
American Mathematical Society