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Mathematics of Computation
Mathematics of Computation
ISSN 1088-6842(online) ISSN 0025-5718(print)

Lucasian criteria for the primality of $ N=h\cdot 2\sp{n} -1$


Author: Hans Riesel
Journal: Math. Comp. 23 (1969), 869-875
MSC: Primary 10.08
MathSciNet review: 0262163
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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.


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DOI: http://dx.doi.org/10.1090/S0025-5718-1969-0262163-1
PII: S 0025-5718(1969)0262163-1
Article copyright: © Copyright 1969 American Mathematical Society