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A generalised Lucasian primality test

Published online by Cambridge University Press:  17 April 2009

Zhenxiang Zhang ,
Weiping Zhou  and
Xianbei Liu
Zhenxiang Zhang
Affiliation:
Department of Mathematics, Anhui Normal University, Wuhu 241000, Anhui, Peoples Republic of China, e-mail: zhangzhx@mail.wh.ah.cn
Weiping Zhou
Affiliation:
Department of Mathematics, Anqing Teachers College, Anqing 246011, Anhui, Peoples Republic of China, e-mail: pingzi212@163.com
Xianbei Liu
Affiliation:
Department of Applied Mathematics, Anhui University of Finance & Economics, Bengbu 233041, Anhui, Peoples Republic of China, e-mail: liuxianbei82@163.com
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We present a primality test for numbers of the form Mh, n = h·2n ±1 (in particular with h divisible by 15), which generalises Berrizbeitia and Berry's test for such numbers with h ≢ 0 mod 5. With our generalised test, the primality of such a number Mh, n can be proved by means of a Lucas sequence with a seed determined by h and πq — primary irreducible divisor of a prime q ≡ 1 mod 4. We call the prime q a judge of the number Mh, n. We prescribe a sequence S of 48 primes ≡ 1 mod 4 in the interval [13, 2593] such that, for all odd h = 15t < 108 and for all n < 7.3 1011, each number Mh, n has a judge q in . Comparisons with Bosma's explicit primality criteria in “a well-defined finite sense” for the case h = 3t < 105 are given.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 74 , Issue 3 , December 2006 , pp. 419 - 441
Copyright
Copyright © Australian Mathematical Society 2006

References

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