Lucasian criteria for the primality of $N=h\cdot 2^{n} -1$
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- by Hans Riesel PDF
- Math. Comp. 23 (1969), 869-875 Request permission
Abstract:
Let $vi = v_{i - 1}^2 - 2$ with ${v_0}$ given. If ${v_{n - 2}} \equiv 0(\bmod N)$ is a necessary and sufficient criterion that $N = h \cdot {2^n} - 1$ be prime, this is called a Lucasian criterion for the primality of $N$. Many such criteria are known, but the case $h = 3A$ has not been treated in full generality earlier. A theorem is proved that (by aid of computer) enables the effective determination of suitable numbers ${v_0}$ for any given $N$, if $h < {2^n}$. The method is used on all $N$ in the domain $h = 3(6)105,n \leqq 1000$. The Lucasian criteria thus constructed are applied, and all primes $N = h \cdot {2^n} - 1$ in the domain are tabulated.References
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E. L. Ince, Cycles of Reduced Ideals in Quadratic Fields, British Association for the Advancement of Science, London, 1934.
D. H. Lehmer, βAn extended theory of Lucasβ functions,β Ann. of Math., v. 31, 1930, pp. 419β448.
- Edouard Lucas, Theorie des Fonctions Numeriques Simplement Periodiques. [Continued], Amer. J. Math. 1 (1878), no.Β 3, 197β240 (French). MR 1505164, DOI 10.2307/2369311
- Hans Riesel, A note on the prime numbers of the forms $N=(6a+1)2^{2n-1}-1$ and $M=(6a-1)2^{2n}-1$, Ark. Mat. 3 (1956), 245β253. MR 76793, DOI 10.1007/BF02589411
- H. C. Williams and C. R. Zarnke, A report on prime numbers of the forms $M=(6a+1)2^{2m-1}-1$ and $M^{\prime } =(6a-1)2^{2m}-1$, Math. Comp. 22 (1968), 420β422. MR 227095, DOI 10.1090/S0025-5718-1968-0227095-2
- Raphael M. Robinson, A report on primes of the form $k\cdot 2^{n}+1$ and on factors of Fermat numbers, Proc. Amer. Math. Soc. 9 (1958), 673β681. MR 96614, DOI 10.1090/S0002-9939-1958-0096614-7
Additional Information
- © Copyright 1969 American Mathematical Society
- Journal: Math. Comp. 23 (1969), 869-875
- MSC: Primary 10.08
- DOI: https://doi.org/10.1090/S0025-5718-1969-0262163-1
- MathSciNet review: 0262163