@MISC{Zhang09onthe, author = {Shaohua Zhang}, title = {On the Infinitude of Some Special Kinds of Primes }, year = {2009} }

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Abstract

The aim of this paper is to try to establish a generic model for the problem that several multivariable number-theoretic functions represent simultaneously primes for infinitely many integral points. More concretely, we introduced briefly the research background–the history and current situation–from Euclid’s second theorem to Green-Tao theorem. We analyzed some equivalent necessary conditions that irreducible univariable polynomials with integral coefficients represent infinitely many primes, found new necessary conditions which perhaps imply that there are only finitely many Fermat primes, generalized Euler’s function, the prime-counting function and Schinzel-Sierpinski’s Conjecture and so on, obtained an analogy of the Chinese Remainder Theorem. By proposed obtrusively several conjectures, we gave a new way for determining the existence of some special kinds of primes. Finally, we proposed sufficient and necessary conditions that several multivariable number-theoretic functions represent simultaneously primes for infinitely many integral points. Nevertheless, this is only a beginning and it miles to go. We hope that number theorists consider further it.