A generalised Lucasian primality test

Abstract

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.