Abstract. A long-standing and almost folkloric conjecture is that the primes contain arbitrarily long arithmetic progressions. Until recently, the only progress on this conjecture was due to van der Corput, who showed in 1939 that there are infinitely many triples of primes in arithmetic progression. In an amazing fusion of methods from analytic number theory and ergodic theory, Ben Green and Terence Tao showed that for any positive integer k, there exist infinitely many arithmetic progressions of length k consisting only of prime numbers. This is an introduction to some of the ideas in the proof, concentrating on the connections to ergodic theory. 1. Background For hundreds of years, mathematicians have made conjectures about patterns in the primes: one of the simplest to state is that the primes contain arbitrarily long arithmetic progressions. It is not clear exactly when this conjecture was first formalized, but as early as 1770 Lagrange and Waring studied the problem of how large the common difference of an arithmetic progression of k primes must be. A

Abstract. Consider the pseudorandom number generator un ≡ u e n−1 (mod m), 0 ≤ un ≤ m − 1, n =1, 2,..., where we are given the modulus m, the initial value u0 = ϑ and the exponent e. One case of particular interest is when the modulus m is of the form pl, where p, l are different primes of the same magnitude. It is known from work of the first and third authors that for moduli m = pl, if the period of the sequence (un) exceeds m3/4+ε, then the sequence is uniformly distributed. We show rigorously that for almost all choices of p, l it is the case that for almost all choices of ϑ, e, the period of the power generator exceeds (pl) 1−ε. And so, in this case, the power generator is uniformly distributed. We also give some other cryptographic applications, namely, to rulingout the cycling attack on the RSA cryptosystem and to so-called time-release crypto. The principal tool is an estimate related to the Carmichael function λ(m), the size of the largest cyclic subgroup of the multiplicative group of residues modulo m. In particular, we show that for any ∆ ≥ (log log N) 3,wehave λ(m) ≥ N exp(−∆) for all integers m with 1 ≤ m ≤ N, apartfromatmost N exp −0.69 ( ∆ log ∆) 1/3) exceptions. 1.

“It is evident that the primes are randomly distributed but, unfortunately, we don’t know what ‘random ’ means. ” — R. C. Vaughan (February 1990). After the first world war, Cramér began studying the distribution of prime numbers, guided by Riesz and Mittag-Leffler. His works then, and later in the midthirties, have had a profound influence on the way mathematicians think about the distribution of prime numbers. In this article, we shall focus on how Cramér’s ideas have directed and motivated research ever since. One can only fully appreciate the significance of Cramér’s contributions by viewing his work in the appropriate historical context. We shall begin our discussion with the ideas of the ancient Greeks, Euclid and Eratosthenes. Then we leap in time to the nineteenth century, to the computations and heuristics of Legendre and Gauss, the extraordinarily analytic insights of Dirichlet and Riemann, and the crowning glory of these ideas, the proof the “Prime Number Theorem ” by Hadamard and de la Vallée Poussin in 1896. We pick up again in the 1920’s with the questions asked by Hardy and Littlewood,

Abstract. In this expository article, we describe the recent approach, motivated by ergodic theory, towards detecting arithmetic patterns in the primes, and in particular establishing in [26] that the primes contain arbitrarily long arithmetic progressions. One of the driving philosophies is to identify precisely what the obstructions could be that prevent the primes (or any other set) from behaving “randomly”, and then either show that the obstructions do not actually occur, or else convert the obstructions into usable structural information on the primes. 1.

this paper we are primarily interested in further developing the theory of quadratic polynomials for which many of the small values are prime (rather than "all" as in Rabinowitsch's result). It is well-known (see [5]) that if the class number of some imaginary quadratic field with large discriminant is one then we will have an egregious counterexample to the Generalized Riemann Hypothesis (that is, a zero of the associated Dirichlet L-function which is very close to 1). Thus Rabinowitsch's result can be informally stated as "n

For S a set of positive integers, and k and r fixed positive integers, denote by f(S,k;r) the least positive integer n (if it exists) such that within every r-coloring of {1,2,...,n} there must be a monochromatic sequence {x1,x2,...,xk} with xi − xi−1 ∈ S for 2 ≤ i ≤ k. We consider the existence of f(S,k;r) for various choices of S, as well as upper and lower bounds on this function. In particular, we show that this function exists for all k if S is an odd translate of the set of primes and r = 2. 1

. We outline some cryptographic applications of the recent results of the authors about small values of the Carmichael function and the period of the power generator of pseudorandom numbers. Namely, we show rigorously that almost all randomly selected RSA moduli are safe against the so-called cycling attack and we also provide some arguments in support of the reliability of the timed-release crypto scheme, which has recently been proposed by R. L. Rivest, A. Shamir and D. A. Wagner. 1. Introduction For an integer n # 1 we define the Carmichael function #(n) as the largest possible order of elements of the unit group in the residue ring modulo n. More explicitly, for a prime power p k we write # p k = p k-1 (p - 1), if p # 3 or k # 2; 2 k-2 , if p = 2 and k # 3; and finally, #(n) = lcm # p k1 1 , . . . , # p k# # , where n = p k1 1 . . . p k# # is the prime number factorization of n. Various upper and lower bounds for #(n) have been...

We generalize conjectures of Hardy and Littlewood concerning the density of twin primes and k-tuples of primes to arbitrary algebraic number fields. In one of their great Partitio Numerorum papers [7], Hardy and Littlewood advance a number of conjectures involving the density of pairs and k-tuples of primes separated by fixed gaps. For example, if d is even, we define P d (x) = |{0 < n < x : n, n + d are both prime}|. They conjecture both that lim x## P d (x) P 2 (x) = # odd p|d p - 1 p - 2 and that P 2 (x) is asymptotic to 2 # p>2 # 1 - 1 (p - 1) 2 # # x 2 dy (log y) 2 . We will refer to the first equation as the "relative conjecture" and the second as the "absolute conjecture." There has been much numerical verification of these conjectures, and many attempts at proofs. Balog [1] proves a result that implies that the conjectures are true "on average," where the average is taken over the possible shapes of the k-tuples. Golubev [6] compares these conjectures with provable analogous limit results for patterns of numbers prime to n. Turan [18] relates such theorems to zeroes of the #-function, using the large sieve rather than Hardy and Littlewood's circle method. There are also many generalizations to specific fields. Most of those generalizations use "Conjecture H" of Sierpinski and Schinzel [14,15]. For example, Sierpinski [17] shows that Conjecture H implies the existence of infinitely many prime Gaussian integers di#ering by 2. Bateman and Horn [2,3] quote a quantitative form of Conjecture H which allows them to estimate the density of rational twin primes. Shanks [16] numerically verifies that the density of prime pairs of the form a + i, a + 2 + i in the Gaussian integers matches that of the quantitative form of Conjecture H. Rieger ...

and that there are infinitely many primes in any arithmetic progression a, a + d, a + 2d,... provided gcd(a, d) = 1andd ≥ 1. If we ask slightly more involved questions, such as whether there exist infinitely many primes of the form n 2 + 1, or infinitely many pairs of primes of the form p, p + 2, then these questions are open,

The first records of studies of prime numbers come from the Ancient Greeks. Euclid proved that there are an infinite number of primes as well as the Fundamental Theorem of Arithmetic which states that every natural number greater than 1 can be written as a unique product of prime numbers. Since the Greeks, prime numbers have been found to apply to more than just