We show how a polynomial-time prover can commit to an arbitrary finite set S of strings so that, later on, he can, for any string x, reveal with a proof whetherÜËorÜ�Ë, without revealing any knowledge beyond the verity of these membership assertions. Our method is non interactive. Given a public random string, the prover commits to a set by simply posting a short and easily computable message. After that, each time it wants to prove whether a given element is in the set, it simply posts another short and easily computable proof, whose correctness can be verified by any one against the public random string. Our scheme is very efficient; no reasonable prior way to achieve our desiderata existed. Our new primitive immediately extends to providing zero-knowledge “databases.”

We present an unconditional deterministic polynomial-time algorithm that determines whether an input number is prime or composite. 1

Dedicated to the memory of W. ‘Red ’ Alford, friend and colleague Abstract. “The problem of distinguishing prime numbers from composite numbers, and of resolving the latter into their prime factors is known to be one of the most important and useful in arithmetic. It has engaged the industry and wisdom of ancient and modern geometers to such an extent that it would be superfluous to discuss the problem at length. Nevertheless we must confess that all methods that have been proposed thus far are either restricted to very special cases or are so laborious and difficult that even for numbers that do not exceed the limits of tables constructed by estimable men, they try the patience of even the practiced calculator. And these methods do not apply at all to larger numbers... It frequently happens that the trained calculator will be sufficiently rewarded by reducing large numbers to their factors so that it will compensate for the time spent. Further, the dignity of the science itself seems to require that every possible means be explored for the solution of a problem so elegant and so celebrated... It is in the nature of the problem

Dedicated to the memory of W. ‘Red ’ Alford, friend and colleague Abstract. “The problem of distinguishing prime numbers from composite numbers, and of resolving the latter into their prime factors is known to be one of the most important and useful in arithmetic. It has engaged the industry and wisdom of ancient and modern geometers to such an extent that it would be superfluous to discuss the problem at length. Nevertheless we must confess that all methods that have been proposed thus far are either restricted to very special cases or are so laborious and difficult that even for numbers that do not exceed the limits of tables constructed by estimable men, they try the patience of even the practiced calculator. And these methods do not apply at all to larger numbers... It frequently happens that the trained calculator will be sufficiently rewarded by reducing large numbers to their factors so that it will compensate for the time spent. Further, the dignity of the science itself seems to require that every possible means be explored for the solution of a problem so elegant and so celebrated... It is in the nature of the problem

Let P denote the set of prime numbers, and let P(n) denote the largest prime factor of an integer n> 1. We show that, for every real number 32/17 < η < (4 + 3 √ 2)/4, there exists a constant c(η)> 1 such that for every integer a � = 0, the set � p ∈ P: p = P(q − a) for some prime q with p η < q < c(η) p η � has relative asymptotic density one in the set of all prime numbers. Moreover, in the range 2 ≤ η < (4+3 √ 2)/4, one can take c(η) = 1+ε for any fixed ε> 0. In particular, our results imply that for every real number 0.486 ≤ ϑ ≤ 0.531, the relation P(q − a) ≍ q ϑ holds for infinitely many primes q. We use this result to derive a lower bound on the number of distinct prime divisors of the value of the Carmichael function taken on a product of shifted primes. Finally, we study iterates of the map q ↦ → P(q − a) for a> 0, and show that for infinitely many primes q, this map can be iterated at least (log log q) 1+o(1) times before it terminates. 1.

Abstract. Brizolis asked the question: does every prime p have a pair (g, h) such that h is a fixed point for the discrete logarithm with base g? The first author previously extended this question to ask about not only fixed points but also two-cycles, and gave heuristics (building on work of Zhang, Cobeli, Zaharescu, Campbell, and Pomerance) for estimating the number of such pairs given certain conditions on g and h. In this paper we extend these heuristics and prove results for some of them, building again on the aforementioned work. We also make some new conjectures and prove some average versions of the results. 1. Introduction and Statement

This is an exposition of the recent paper PRIMES is in P by Manindra Agrawal, Neeraj Kayal, and Nitin Saxena, which for the first time gives a polynomial-time algorithm for determining whether a given integer n is prime or composite. We claim no originality in this paper, but there are a few minor variants here of the

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: For a given polynomial f we use `local' methods to find exponents k for which there are no non--trivial integer solutions x 1 ; x 2 ; : : : ; xn to the Diophantine equation f(x k 1 ; x k 2 ; : : : ; x k n ) = 0 1. Introduction For a given polynomial f(X 1 ; X 2 ; : : : ; Xn ) 2 Z[X 1 ; X 2 ; : : : ; Xn ] we shall investigate the set T (f) of exponents k for which the Diophantine equation (1) f(x k 1 ; x k 2 ; : : : ; x k n ) = 0 has solutions in non--zero integers x 1 ; x 2 ; : : : ; xn . For homogenous diagonal f of degree one, Davenport and Lewis showed that k 2 T (f) whenever (n \Gamma 1) 1=2 k 18; however, Ankeny and Erdos [AE] showed that T (f) has zero density in the set of all positive integers provided that all distinct subsets of the set of coefficients of f have different sums. For general polynomials f , Ribenboim [R] showed that certain values of k cannot belong to T (f ), and the result of Ankeny and Erdos shows that T (f) has zero density, under the sam...

These notes contain a description and correctness proof of the deterministic polynomial-time primality testing algorithm of Agrawal, Kayal, and Saxena. Some background from number theory and algebra is given in Section 4. 1 A polynomial identity for prime numbers Theorem 1.1 Let n 2 and a 0 be integers. 1. If n is a prime number, then in the ring Z n [x].