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, “An extended theory of Lucas’ functions,” Ann. of Math., v. 31, 1930, pp. 419–448.
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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