| On the Least Prime in Certain Arithmetic Progressions (2008) | |||||||||||||
Abstract | |||||||||||||
| We find infinitely many pairs of coprime integers, a and q, such that the least prime j a (mod q) is unusually large. In so doing we also consider the question of approximating rationals by other rationals with smaller and coprime denominators. * The second author is partly supported by an N.S.F. grant 1. Introduction For any x ? x 0 and for any positive valued function g(x) define R(x) = e fl log x log 2 x log 4 x=(log 3 x) 2 ; L(x) = exp(log x log 3 x= log 2 x) and E g (x) = exp \Gamma log x=(log 2 x) g(x) \Delta : Here log k x is the k-fold iterated logarithm, fl is Euler's constant, and x 0 is chosen large enough so that log 4 x 0 ? 1. The usual method used to find large gaps between successive prime numbers is to construct a long sequence S of consecutive integers, each of which has a "small" prime factor (so that they cannot be prime); then, the gap between the largest prime before S and the next, is at least as long as S. Similarly if one wishes to find an arithm... | |||||||||||||
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