The Fueter–Pólya theorem, first proved by Rudolf Fueter and George Pólya, states that the only quadratic pairing functions are the Cantor polynomials.
Contents
Introduction
In 1873, Georg Cantor showed that the so-called Cantor polynomial
is a bijective mapping from
Fueter was investigating whether there are other quadratic polynomials with this property, and concluded that this is not the case assuming
Statement
If
or
Proof
The original proof is surprisingly difficult, using the Lindemann–Weierstrass theorem to prove the transcendence of
Fueter–Pólya conjecture
The theorem states that the Cantor polynomial is the only quadratic paring polynomial of
Higher dimensions
The generalization of the Cantor polynomial in higher dimensions is as follows:
The sum of these binomial coefficients yields a polynomial of degree