Matematické Fórum

Skrytý text:
For the equation gives . We can easily prove, tak if for some positive integer, than there exists for which :



So we know, that holds for infinitely many numbers. Assume that p is polynomial of degree d, . We know, that there exists d+1 different numbers , for which



This is a system of d+1 linear equations of d+1 variables . Since we know, that Vandermonde matrix has non-zero determinant ( are different numbers), this system has unique solution. It is obvious, that this solution is for and . The polynomial p has to be of the form . It is trivial to verify that p(x)=x is really the solution of our problem.