一道超难的数论题,最简分数m/n=1+1/2+1/3+……+1/2010,试证明m为质数2011的倍数.

Question

一道超难的数论题,
最简分数m/n=1+1/2+1/3+……+1/2010,试证明m为质数2011的倍数.

Answer 1

设Un={1,2,...,n},v(n,k)=sum{r1*r2*...*rk|r1,r2,...,rk∈Un,ri≠rj}再设f(n,x)=(x+1)(x+2)...(x+n),则f(n,x)=x^n+v(n,1)*x^(n-1)+...+v(n,n-1)*x+v(n,n)引理：对于任意奇数n,(n+1)(n+2)...(2n-1) = (n-1)!(mod n^2...