By Hardy G.H., Wright E.M.

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2 (The extended Riemann hypothesis (ERH)). Let X be an arbitrary Dirichlet character. Then the zeros of L(s, X) in the region Re( s) > 0 lie on the vertical line Re( s) = ~. We note that an even more general hypothesis, the generalized Riemann hypothesis (GRH) is relevant for more general algebraic domains, but we limit the scope of our discussion to the ERH above. 2 is of fundamental importance also in computational number theory. For example, one has the following conditional theorem. 5. Assume the ERH holds.

To show how hard these investigators must have searched, the prime divisor Cosgrave found is itself currently one of the dozen or so largest known primes. Similar efforts reported recently by [Dubner and Callot 2000] include K. Herranen's generalized Fermat prime and S. Scott's gargantuan prime Chapter 1 PRIMES! 28 A compendium of numerical results on Fermat numbers is available at [Keller 1999]. 2 that there are infinitely many primes: Since the Fermat numbers are odd and the product of Fa, F1 , ...

A heuristic suggested by H. 13) 2n ' where b is the current limit on the possible prime factors of Fn. If nothing is known about possible factors, one might use the smallest possible prime factor b = 3 . 2n +2 + 1 for the numerator calculation, giving a rough a priori probability of n/2 n that Fn is prime. ) It is from such a probabilistic perspective that Fermat's guess looms as ill-fated as can be. 3 Certain presumably rare primes There are interesting classes of presumably rare primes. We say "presumably" because little is known in the way of rigorous density bounds, yet empirical evidence and heuristic arguments suggest relative rarity.