Originally Posted by

**Sandwich** Hello,

I look for ALL polynomials with the two following qualities:

1) All co-efficients of p are in N

2) An function f exists with f(f(f(x)))=p(n) for all n in N. (The domain of f is N and the co-domain of f is N, too)

N is the set of all natural numbers WITH 0: N={0, 1, 2, 3, 4, ...}

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I have no idea how to solve this problem, so I started to find examples for such polynomials p.

1. Obviously p=x^{3 * z} for all z ∈ N is such an polynomial.

2. All q=x^{z} with z ∈ N and 3|z are NOT such polynomials p.

3. I calculated f(f(f(x))) with f(x)=x^2+x. Hence p(x)=x^8+4*x^7+8*x^6+10*x^5+9*x^4+6*x^3+3*x^2+x is an polynomial, which I look for.

4. I calculated f(f(f(x))) with f(x)=x^2+2*x. Hence p(x)=x^8+8*x^7+28*x^6+56*x^5+70*x^5+56*x^3+28*x^2+ 8x is an polynomial, which I look for.

So, how do I find all other polynomials p ?

Sanwich