Fueter–Pólya theorem

The Fueter–Pólya theorem, first proved by Rudolf Fueter and George Pólya, states that the only quadratic pairing functions are the Cantor polynomials.

In 1873, Georg Cantor showed that the so-called Cantor polynomial

is a bijective mapping from ${\displaystyle \mathbb {N} ^{2}}$ to ${\displaystyle \mathbb {N} }$. 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 ${\displaystyle P(0,0)=0}$. He then wrote to Pólya, who showed the theorem does not require this condition.

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