منابع مشابه
An elliptic curve test for Mersenne primes
Let l ≥ 3 be a prime, and let p = 2 − 1 be the corresponding Mersenne number. The Lucas-Lehmer test for the primality of p goes as follows. Define the sequence of integers xk by the recursion x0 = 4, xk = x 2 k−1 − 2. Then p is a prime if and only if each xk is relatively prime to p, for 0 ≤ k ≤ l − 3, and gcd(xl−2, p) > 1. We show, in the first section, that this test is based on the successiv...
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The Biquadratic Reciprocity Law is used to produce a deterministic primality test for Gaussian Mersenne norms which is analogous to the Lucas–Lehmer test for Mersenne numbers. It is shown that the proposed test could not have been obtained from the Quadratic Reciprocity Law and Proth’s Theorem. Other properties of Gaussian Mersenne norms that contribute to the search for large primes are given....
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We introduce a new class of pseudoprimes-so called ”overpseudoprimes” which is a special subclass of super-Poulet pseudoprimes. Denoting via h(n) the multiplicative order of 2 modulo n,we show that odd number n is overpseudoprime if and only if the value of h(n) is invariant of all divisors d > 1 of n. In particular, we prove that all composite Mersenne numbers 2 − 1, where p is prime, and squa...
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ژورنال
عنوان ژورنال: Journal of Number Theory
سال: 2005
ISSN: 0022-314X
DOI: 10.1016/j.jnt.2003.11.011