[1/2] I tweeted recently about Cantor’s famous polynomial bijection from all pairs of non-negative integers to the set of all non-negative integers.

Now, it’s easy to find bijections from the pairs of integers to the non-negative integers, like the one pictured below:

[2/2] It’s also easy to find bijections from the pairs of integers to the integers.

But AFAIK, it remains an open problem whether there’s even an injective *polynomial* [one that never repeats the same value] from the pairs of integers to the integers, let alone a bijection.

Thanks to everyone who pointed out the Maths Overflow discussion on the closely related question for rationals:

https://mathoverflow.net/questions/21003/polynomial-bijection-from-mathbb-q-times-mathbb-q-to-mathbb-q

and the quote:

“Don Zagier has speculated that a polynomial as simple as x^7+3y^7 might already be an example.”

It’s still open, of course…

… but here are a few comments on the general idea of using:

P(x,y) = x^a + b y^a

First, a must be an *odd* number, or we’d have:

P(x,y) = P(–x,–y)

Second, b=1 is no good, because that gives:

P(–1,1)=P(1,–1)=0

And b=2 is no good, because that gives:

P(–1,0)=P(1,–1)=–1.

What about a? a=3 is no good, because given integers x≠0, y≠0, we get the same value for P at the distinct integer points:

(x(x^3 + 2by^3), –y(2x^3 + by^3))

and:

(x(x^3 – by^3), y(x^3 – by^3))

So we need a to be at least 5, and b to be at least 3.