The Study About Fermat Number

The problem about Fermat numbers is an unsolved and well-known problem of number theory in the world. Fermat,P.de put forward a conjecture: The numbers which satisfied F_n = 2²ⁿ +1 are all prime. But he didn't give out a complete proof.In 1732,the famous mathematical master Euler found that F₅ =641·6700417 which meaned that F₅ was a composite number. So the Fermat's conjecture was wrong.After that people researched about more and more Fermat numbers. So far, Fermat prime numbers have not been found any more ,except the five Fermat prime numbers which were found by Fermat.P.de .So Hardy and Wright gave out a discussion which was reasonable and rich in enlightment. They thought that the Fermat prime numbers were finite. Selfridge supported the following conjecture: All the other Fermat numbers are composite numbers .The author in this paper extended DengGao Zeng's and YiYuan Mei's conclusions and obtained the conclusions (1), (6). In the base of Professor MaoHua Le, A.Grytczuk and M. Wojtowicz,the author in this paper made farther exploration on the problem about the lower bound of the greatest prime factor of the Fermat numbers and got the conclusion (7).Besides that the author also get the other conclusions about Fermat numbers.