摘  要:本文从素数的基本概念出发，介绍了素数有无穷多个的性质，并且给出了一些伟大的数学家给出的关于素数有无穷多个的几个新证明，如欧几里德、歌德巴赫、欧拉等数学家对素数无穷多的的独特的证明，至今被后人称道。千百年人们对素数作了艰难的探索并得出了一些杰出的成果，但关于素数的一些猜想至今仍未得到证明，可见这个领域还是较为有讨论价值的.本文将展示前人的一些伟大成果，旨在使读者对素数、素数无穷多有一个较全面的了解，并争取在研究过程中有新的突破。

关键词：素数； 无穷多； 反证法；N2+1形素数； 4n+1形素数。

Abstract:This paper from the basic concept of prime, introduces the properties of an infinite number of primes, and gives some great mathematicians give about an infinite number of primes, several new proofs, such as Euclid, GeDeBaHe, euler infinite number of primes such mathematician the unique proof, so far to be remembered praised. Thousands of people to make the tough exploration prime and obtained some outstanding achievements, but some guesses about primes have so far not been proofed, visible in this area or more. This paper will show some of the great achievements, predecessors to primes, aims to make readers have a prime infinite, and more comprehensive understanding in the process of research for a new breakthrough.

Key words：prime; infinite; proof by contradiction; N2+1Figured primes;  4n+1Figured primes.