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E31. A packing polynomial

Funny how one comes across math problems. I was reading my Reed College alumni magazine and came across an article about Maddie Grant, class of '15, whose undergraduate thesis is apparently a substantial generalization of a 1923 result by Fueter and Polya. Feuter and Polya evidently discovered a polynomial function that maps the non-negative integers one-to-one and onto the pairs of non-negative integers. That is, they found a simple formula to express the inverse Cantor's "diagonal" mapping of pairs of non-negative integers to non-negative integers. Their function – a so-called packing polynomial – is given by

f(x,y) = (1/2)[(x + y)2 + x + 3y],
where x and y are non-negative integers.

The proof that this function is one-to-one and onto the non-negative integers is actually pretty elementary if you look at it the right way. I will print a hint as a comment in a few days.


Peter Ash said...

Hint: Let x + y = n. Then
f(x,y) = (1/2)(n)(n + 1) + y = (1 + ... + n) + y.

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