On Primes Recognizable in Deterministic Polynomial Time
In this paper we present several algorithms that can find proofs of primality in deterministic polynomial time for some primes. In particular we show this for any prime p for which the complete prime factorization of p − 1 is given. We can also show this when a completely factored divisor of p − 1 is given that exceeds p l/4+ε. And we can show this if p − 1 has a factor F exceeding p ε with the property that every prime factor of F is at most (log p)2/ε. Finally, we present a deterministic polynomial time algorithm that will prove prime more than x 1-ε primes up to x, The key tool we use is the idea of a smooth number, that is, a number with only small prime factors. We show an inequality for their distribution that perhaps has independent interest.
KeywordsFactorization Method Binary Search Deterministic Algorithm Primitive Root Nontrivial Factorization
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