Biquadratic reciprocity and a Lucasian primality test

Let {s k , k > 0} be the sequence defined from a given initial value, the seed, so, by the recurrence s k+1 = s 2 k - 2,k > 0. Then, for a suitable seed so, the number M h,n = h.2 n - 1 (where h < 2 n is odd) is prime iff s n-2 ≡ 0 mod M h,n . In general so depends both on h and on n. We describe a slight modification of this test which determines primality of numbers h.2 n ±1 with a seed which depends only on h, provided h ≢ 0 mod 5. In particular, when h = 4 m - 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 2 k ± 1, Math. Comp. 61 (1993), 97-109, needs infinitely many seeds. The proof of validity uses biquadratic reciprocity.