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在“科学”第32卷第11期上(327页),作者已证明下列二定理:定理一如4n+1为质数(n 为正整数),则 n~n(?)1(mod 4n+1).定理二如4n+1为质数(n 为正整数),则 4~n(?)(-1)~n(mod 4n+1).设若此二定理之逆定理可以成立,则凡作4n+1形式之数,即可藉以判定其是否质数.惜乎在事实上虽有许多实例可以成立,然欲证之,殊不可能,盖此逆定理不能普遍成立故也。兹举一例可以见之:如4n+1=2~(2~m)+1,则无论4n+1是否质数,必有 n~n(?)1(mod 4n+1)
In “Science” Volume 32, Number 11 (327), the author has proved the following second theorem: Theorem is as 4n + 1 as a prime number (n is a positive integer), then n ~ n (?) 1 (mod 4n + 1). Theorem Two If 4n+1 is a prime number (n is a positive integer), then 4~n(?)(-1)~n(mod 4n+1). If the inverse theorem of the second theorem can be established, then The number of 4n+1 forms can be used to determine whether it is a prime number. It is regrettable that although there are many instances in fact that can be established, it is impossible to prove it, and it is not possible to cover this inverse theorem. Here is an example to see: if 4n+1=2~(2~m)+1, then whether 4n+1 is a prime number, there must be n~n(?)1(mod 4n+1)