Abstract:
We describe a very general abstract form of sieve based on a large sieve inequality which generalizes both the classical sieve inequality of Montgomery (and its higher-dimensional variants), and our recent sieve for Frobenius over function fields. The general framework suggests new applications. We get some first results on the number of prime divisors of ``most'' elements of an elliptic divisibility sequence, and we develop in some detail ``probabilistic'' sieves for random walks on arithmetic groups, e.g., estimating the probability of finding a reducible characteristic polynomial at some step of a random walk on SL(n,Z). In addition to the sieve principle, the applications depend on bounds for a large sieve constant. To prove such bounds involves a variety of deep results, including Property (T) or expanding properties of Cayley graphs, and the Riemann Hypothesis over finite fields. It seems likely that this sieve can have further applications.

Abstract:
A sieve is constructed for ordinary twin primes of the form 6m+/-1 that are characterized by their twin rank m. It has no parity problem. Non-rank numbers are identified and counted using odd primes p>=5. Twin- and non-ranks make up the set of positive integers. Regularities of non-ranks allow gathering information on them to obtain a Legendre-type formula for the number of twin-ranks near primorial arguments.

Abstract:
Here we demonstrate a sieve for analysing primes and their composites, using equivalence classes based on the modulo 6 return value as applied to the Natural numbers. Five features of this 'Hexile' sieve are reviewed. The first aspect, is that it narrows the search for primes to one-third of the Natural numbers. The second feature is that we can obtain from the equivalence class formulae, a property of its diophantine equations to distinguish between primes and composites resulting from multiplication of these primes. Thirdly we can from these diophantine formulations ascribe a non-random occurence to not only the composites in the two equivalence classes but by default and as a consequence : non-randomness of occurence to the resident primes. Fourthly we develop a theoretical basis for sieving primes. Of final mention is that the diophantine equations allows another route to a prime counting function using combinatorics or numerical analysis.

Abstract:
Using the properties of the table sieve, we can determine whether all given number, positive integer G, is a prime and whether it is possible to factor it out.

Abstract:
We derive, for all prime moduli p except those in a very thin set, an upper bound for the least prime primitive root (mod p) of order of magnitude a constant power of log p. The improvement over previous results, where the upper bound was log p to an exponent tending to infinity with p, lies in the use of the linear sieve (a particular version called the shifted sieve) rather than Brun's sieve. The same methods allow us to rederive a conditional result of Shoup on the least prime primitive root (mod p) for all prime moduli p, assuming the generalized Riemann hypothesis. We also extend both results to composite moduli q, where the analogue of a primitive root is an element of maximal multiplicative order (mod q).

Abstract:
We give a more comrehensive treatment of Chen's double sieve and improve related constants in Goldbach's conjecture and the twin prime problem.

Abstract:
Using the sieve, we show that there are infinitely many Carmichael numbers whose prime factors all have the form $p = 1 + a^2 + b^2$ with $a,b \in{\mathbb Z}$.

Abstract:
We present a new sieve that allows us to find the prime numbers by using only regular patterns and, more importantly, avoiding any duplication of elements between them.

Abstract:
The Sieve of Eratosthenes has been recently extended by excluding the multiples of 2, 3, and 5 from the initial set, and finding the additive rules that give the positions of the multiples of the remaining primes. We generalize these results. For a given k we let the initial set Sk consists of natural numbers relatively prime to the first k primes, and find the rules governing the positions of the multiples of the remaining elements.

Abstract:
Using a sieve procedure akin to the sieve of Eratosthenes we show how for each prime $p$ to build the corresponding M\"obius prime-function, which in the limit of infinitely large primes becomes identical to the original M\"obius function. Discussing this limit we present two simple proofs of the Prime Number Theorem. In the framework of this approach we give several proofs of the infinitude of primes.