Sieve theory and the distribution of Sample Clauses

Sieve theory and the distribution of primes Recall that sieve techniques, developed by Brun and expanded by others, consider “approximations” to classical questions about the distribution of primes (e.g., the twin prime conjecture and Xxxxxxxx’x conjecture): letting P2 denote the set of integers that are either prime or the product of two primes, it is known that there are infinitely many primes p such that p + 2 is in P2, and it is also known that every sufficiently large even integer is the sum of a prime and an element of P2. Another classical problem concerns primes represented by polynomials. That is, given an irreducible polynomial F (x), is F (n) prime infinitely often, where n ∈ Z? It turns out that there can be local obstructions (e.g., the polynomial F (x) = x2 + x + 2 is always even), but for polynomials without local obstructions, we can follow similar heuristic reasoning as we did with the twin primes, and we conjecture that F (n) is prime infinitely often. Dirichlet’s theorem on primes in arithmetic progressions yields this conjecture in the case deg F (x) = 1, but for no higher degree polynomials is it known. In Chapter 3, we prove the following theorem.