BIQUADRATIC RECIPROCITY AND A LUCASIAN PRIMALITY TEST

Abstract. Let {sk,k ≥ 0} be the sequence defined from a given initial value, the seed, s0, by the recurrence sk+1 = s2 k − 2,k ≥ 0. Then, for a suitable seed s0, thenumberMh,n = h · 2n − 1(whereh<2n is odd) is prime iff sn−2 ≡ 0modMh,n. In general s0 depends both on h and on n. We describe a slight modification of this test which determines primality of numbers h·2n ±1 with a seed which depends only on h, providedh� ≡ 0 mod 5. In particular, when h =4m − 1, m odd, we have a test with a single seed depending only on h, in contrast with the unmodified test, which, as proved by W. Bosma in Explicit primality criteria for h · 2k ± 1, Math. Comp. 61 (1993), 97–109, needs infinitely many seeds. The proof of validity uses biquadratic reciprocity. The Lucasian sequence with seed s0 is the sequence {sk} defined from the given initial value s0 by the recurrence sk+1 = s2 k − 2, k ≥ 0. A Lucasian primality test is a primality test involving a Lucasian sequence. The terminology comes from the Lucas-Lehmer test for Mersenne primes (see [4] for historical details): Theorem 1 (Lucas-Lehmer). Let p be an odd prime, and let Mp =2p − 1 be th