Fine distribution of primes

Besides the basic problem of counting primes, there are many interesting questions about what kinds of special primes exist. For instance, when looking over the list of primes, occasionally we will see pairs like (11,13), (17,19), (71,73), (1031,1033). No matter how far we extend the list, there always seems to appear another prime pair of this kind. A pair of primes of the form p, p+2 is called a pair of twin primes. The twin prime conjecture is that infinitely many pairs of prime twins exist. This is still unproved today. It is also unknown whether or not there exist infinitely many primes of the form p=n²+1, although the list in this case also appears unending, e.g. 5=2²+1, 17=4²+1, 37=6²+1, 101=10²+1, etc. Goldbach's conjecture asserts that every even number >2 is the sum of two primes, for instance, 4=2+2, 6=3+3, 8=3+5, 10=3+7, etc. This statement, which has so far proved totally baffling, also describes something about the distribution of primes, in the sense that for the conjecture to be true the primes must be sufficiently uniformly distributed to provide pairs adding up to any even number. One of the closest ``near-misses'' is a theorem of the Russian mathematician Vinogradov that every odd integer greater than 10⁸⁰ is a sum of three primes. Computers large enough to check all the integers less than or equal to 10⁸⁰ unfortunately do not exist yet.