Gaussian Mersenne and Eisenstein Mersenne primes
نویسندگان
چکیده
منابع مشابه
Gaussian Mersenne and Eisenstein Mersenne primes
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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Article history: Received 18 June 2014 Received in revised form 11 June 2015 Accepted 12 June 2015 Available online 22 June 2015 Communicated by Neal Koblitz MSC: primary 11A41, 11A07 secondary 15B99, 05C90
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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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ژورنال
عنوان ژورنال: Mathematics of Computation
سال: 2010
ISSN: 0025-5718
DOI: 10.1090/s0025-5718-10-02324-0