User blog comment:Vel!/Call for math facts/@comment-1605058-20140731145031


 * Elliot-Halberstam conjecture concerns the difference between number of primes in arithmetic progressions and the asymptotic estimate by Dirichlet's theorem and prime number theorem. Surprising connections between this conjecture and distribution of prime constellations has been discovered.


 * Dirichlet's theorem on primes in arithmetic progressions shows that in every arithmetic progression there is infinite number of primes, unless there is a factor larger than 1 common to all elements of progression. No such result is known for any higher order polynomials, yet Bunyakovsky conjecture states that every polynomial satisfying few requirements takes prime values infinitely many times.


 * Twin prime conjecture is vastly generalized by Schinzel's hypothesis H, saying that if we have any finite set of irreducible integral polynomials P_i(x), if their product does not have a fixed divisor dividing each of its values, then there exist infinitely many integer values of x such that all of P_i(x) are prime at once. To derive twin prime conjecture, just use P_1(x)=x, P_2(x)=x+2.