  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, obtained an analogy of
the Chinese Remainder Theorem, generalized Euler's function, the prime-counting
function and Schinzel-Sierpinski's Conjecture and so on. Nevertheless, this is
only a beginning and it miles to go. We hope that number theorists consider
further it.
