Create unique number from 2 numbers

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is there some way to create unique number from 2 positive integer numbers? Result must be unique even for these pairs: 2 and 30, 1 and 15, 4 and 60. In general, if I take 2 random numbers result must be unique(or with very high probability unique)

Thanks a lot

EDIT: calculation is for computer program,so computational complexity is important

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6 Answers

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The sort of function you are looking for is often called a pairing function.

A standard example is the Cantor pairing function $\mathbb{N}\times\mathbb{N}\rightarrow\mathbb{N}$, given by: $\pi(a,b) = \frac{1}{2}(a+b)(a+b+1) + b$.

You can find more information here:

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if the numbers are $a$ and $b$, take $2^a3^b$. This method works for any number of numbers (just take different primes as the bases), and all the numbers are distinct.

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Google pairing function. As I mentioned in the similar question, there are also other pairing functions besides the well-known one due to Cantor. For example, see this "elegant" pairing function, which has the useful property that it orders many expressions by depth.

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For positive integers as arguments and where argument order doesn't matter:

1. Here's an unordered pairing function:

   $<x, y> = x * y + trunc(\frac{(|x - y| - 1)^2}{4}) = <y, x>$

2. For x ≠ y, here's a unique unordered pairing function:

   <x, y> = if x < y: x * (y - 1) + trunc((y - x - 2)^2 / 4) if x > y: (x - 1) * y + trunc((x - y - 2)^2 / 4) = <y, x>

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You could try the function by Matthew Szudzik, which is given as:$$a \geq b ~?~ a*a + a + b : a+b*b$$

In other words: $$(a,b)\mapsto \begin{cases} a^2 + a + b & \text{if } a \geq b\\ a + b^2 & \text{if } a < b \end{cases}$$

I found it here, which is a similar question as this.

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Apologies for resurrecting this ancient question, but I've noticed that there are collisions in the results of the Cantor pairing function.

For example, I've noticed that when a and b are 0.0 and 0.0, or 0.6 and 0.0 the result is the same: 0.48.

Is this function expected to work for non-discrete numbers, or does this have something to do with the fact they the numbers are 0.0 and 0.0 in the former case?

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