A Survey of Results on Primes in Short Intervals

Prime numbers have been a source of fascination for mathematicians since antiquity. The proof that there are infinitely many prime numbers is attributed to Euclid (fourth century B.C.). The basic method of determining all primes less than a given number N is the sieve of Eratosthenes (third century B.C.). Diophantus (third (?) century A.D.) was occupied with finding rational number solutions to equations, extending ancient knowledge from Babylon and India on Pythagorean triples. The books of Diophantus lay lost for ages. It took thousands of years before new aspects of primes were brought into light until chiefly Fermat and Mersenne (c.1640), influenced by Bachet’s (1621) translation into Latin of the extant books of Diophantus, announced various criteria on divisibility by primes, assertions on primes possessing special forms, and solutions to Diophantine equations.