The problem of quickly determining whether a given large integer is prime or composite has been of interest for centuries, if not longer. The past 30 years has seen a great deal of progress, leading up to the recent deterministic, polynomial-time algorithm of Agrawal, Kayal, and Saxena [2]. This new “AKS test ” for the primality of n involves verifying the

Abstract. This note presents the basic mathematical structure of a new integer factorization method based on systems of linear Diophantine equations. The estimated theoretical running time complexities of the corresponding algorithms are encouraging and improve the current ones. The work is presented as a theoretical contribution to the theory of integer factorization.

Abstract. In this paper we prove three conjectures on congruences involving central binomial coefficients or Lucas sequences. Let p be an odd prime and let a be a positive integer. We show that if p ≡ 1 (mod 4) or a> 1 then ⌊ 3 4 pa ⌋ k=0

Abstract. Let p be an odd prime. It is well known that F p p− () 5 0 (mod p) where {Fn} n�0 is the famous Fibonacci sequence and (−) is the Jacobi symbol. In this paper we show that if p ̸ = 5 then we may determine F p p−( 5) mod p3 in the following way: (p−1)/2 k=0

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