The Fueter–Pólya theorem, first proved by Rudolf Fueter and George Pólya, states that the only quadratic polynomial pairing functions are the Cantor polynomials.
In 1873, Georg Cantor showed that the so-called Cantor polynomial[1] is a bijective mapping from
The polynomial given by swapping the variables is also a pairing function.
Fueter was investigating whether there are other quadratic polynomials with this property, and concluded that this is not the case assuming
He then wrote to Pólya, who showed the theorem does not require this condition.
is a real quadratic polynomial in two variables whose restriction to
then it is or The original proof is surprisingly difficult, using the Lindemann–Weierstrass theorem to prove the transcendence of
[3] In 2002, M. A. Vsemirnov published an elementary proof of this result.
The conjecture is that these are the only such pairing polynomials, of any degree.
A generalization of the Cantor polynomial in higher dimensions is as follows:[5] The sum of these binomial coefficients yields a polynomial of degree