On the Infinitude of Some Special Kinds of Primes

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, 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.

The Solution of Fermat’s Two Squares Equation and Its Generalization In Lucas Sequences

“…The proofs of these propositions can be obtained from the results of Theorems 2,4,5,6,and 8,respectively. As there are no solutions for Eq. 12 where (𝑥, 𝑦, 𝑝)∈{(𝐿 𝑖 , 𝐿 𝑗 , 𝐿 𝑘 ),(𝑄 𝑖 , 𝑄 𝑗 , 𝑄 𝑘 ), (𝑀 𝑖 , 𝑀 𝑗 , 𝑀 𝑘 ), (𝑁 𝑖 , 𝑁 𝑗 , 𝑁 𝑘 ), (𝐸 𝑖 , 𝐸 𝑗 , 𝐸 𝑘 )}, it can be easily concluded that Eq.…”

“…Furthermore, Dirichlet proved that there are an infinite number of primes of the form 𝑝 = 𝑎𝑘 + 𝑏 if 𝑔𝑐𝑑(𝑎, 𝑏) = 1, where 𝑎 and 𝑏 are integers, see e.g. 6 . Indeed, this latter result demonstrates that there are an infinite number of primes of the form 𝑝 = 𝑥 2 + 𝑦 2 .…”