Small Gaps between Primes I
نویسنده
چکیده
Finding mathematical proofs for easily observed properties of the distribution of prime numbers is a difficult and often humbling task, at least for the authors of this paper. The twin prime conjecture is a famous example of this, but we are concerned here with the much more modest problem of proving that there are arbitrarily large primes that are “unusually close ” together. Statistically this means we seek consecutive primes whose distance apart is substantially less than the average distance between consecutive primes. Letting pn denote the n prime, then by the prime number theorem the average gaps size pn+1 − pn between consecutive primes is log pn. Thus we define
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