  We introduce a method for showing that there exist prime numbers which are
very close together. The method depends on the level of distribution of primes
in arithmetic progressions. Assuming the Elliott-Halberstam conjecture, we
prove that there are infinitely often primes differing by 16 or less. Even a
much weaker conjecture implies that there are infinitely often primes a bounded
distance apart. Unconditionally, we prove that there exist consecutive primes
which are closer than any arbitrarily small multiple of the average spacing,
that is, \[ \liminf_{n\to \infty} \frac{p_{n+1}-p_n}{\log p_n} =0 .\] This last
result will be considerably improved in a later paper.
