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Abstract. We present algorithms that are deterministic primality tests for a large family of integers, namely, integers n ≡ 1 (mod 4) for which an integer a is given such that the Jacobi symbol ( a n ) = −1, and integers n ≡ −1 (mod 4) for which an integer a is given such that ( a n ) = ( 1−a n ) = −1. The algorithms we present run in 2 − min(k,[2 log log n])Õ ((log n) 6 ) time, where k = ν 2 (n − 1) is the exact power of 2 dividing n − 1 when n ≡ 1 (mod 4) and k = ν 2 (n + 1) if n ≡ −1 (mod 4). The complexity of our algorithms improves up toÕ((log n) 4 ) when k ≥ [2 log log n]. We also give tests for a more general family of numbers and study their complexity.

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Abstract. Let {s k , k ≥ 0} be the sequence defined from a given initial value, the seed, s 0 , by the recurrence s k+1 = s 2 k − 2, k ≥ 0. Then, for a suitable seed s 0 , 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 s 0 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.The Lucasian sequence with seed s 0 is the sequence {s k } defined from the given initial value s 0 by the recurrence Let n, h ∈ N with h odd, h < 2 n , and let M n,h = h · 2 n − 1. The Lucas-Lehmer test generalizes to a Lucasian primality test for M = M h,n as follows: (1) M is prime.For a proof see [3]. The Lucas-Lehmer test is the special case d = 3, α = −1 + √ 3 of this theorem. For then Tr Q( √ 3)/Q (α) = −4 whence (the sign being clearly irrelevant) s 0 = 4. The generalization differs from the original Lucas-Lehmer test in two respects. First, in the generalized test the seed is a rational number, which may not be an integer; this is not a serious difficulty, since, by inverting the denominator mod M , one can replace the rational seed by an integer seed. Second,

We describe probabilistic primality tests applicable to integers whose prime factors are all congruent to 1 mod r where r is a positive integer; r = 2 is the Miller-Rabin test. We show that if ν rounds of our test do not find n = (r + 1) 2 composite, then n is prime with probability of error less than (2r) −ν . Applications are given, first to provide a probabilistic primality test applicable to all integers, and second, to give a test for values of cyclotomic polynomials.