Produce unique number given two integers

Question

Given two integers, $a$ and $b$, I need an operation to produce a third number $c$. This number does not have to be an integer. The restrictions are as follows:

1. $c$ must be unique for the inputs (but it does not have to be reversible).
2. $a$ and $b$ must be interchangeable ($a$ & $b$ = $b$ & $a$)

Initially, the first thing I thought of was simply $a+b$, however naturally that does not fit restriction 1. Then I considered a hash function of some sort, but that doesn't fit 2.

Any thoughts?

Answer 1

How about just $2^a+2^b$? This represents the binary number with $1$s at exactly the $a$-th and $b$-th positions if $a\ne b$, and a single $1$ at the $(a+1)$-th position if $a=b$.

Answer 2

If we restrict $a,b$ to be non-negative integers, we can try $f(a,b) = \dfrac{\max(a,b)(\max(a,b)+1)}{2}+\min(a,b)$.

This satisfies $f(a,b) = f(b,a)$ and grows quadratically with $\max(a,b)$. To help you see the pattern:

$f(0,0) = 0$,

$f(1,0) = 1$, $f(1,1) = 2$,

$f(2,0) = 3$, $f(2,1) = 4$, $f(2,2) = 5$,

$f(3,0) = 6$, $f(3,1) = 7$, $f(3,2) = 8$, $f(3,3) = 9$,

...

If you want to allow any integers $a,b$ then let $g$ be your favorite bijection from $\mathbb{Z}$ to $\mathbb{N}_0$, then let $f(a,b) = \dfrac{\max(g(a),g(b))(\max(g(a),g(b))+1)}{2}+\min(g(a),g(b))$.

One such bijection is $g(n) = \begin{cases} -2n & n \le 0 \\ 2n-1 & n \ge 1\end{cases}$.

Answer 3

I think $a\circ b=(2^a+1)(2^b+1)$ works, if I understand correctly. $a \circ b = b \circ a$ and $a$ and $b$ can be found (up to permutation) from $(2^a+1)(2^b+1)$ (assuming $a$ and $b$ are integers.)

Answer 4

Peter Woolfitt's suggestion ($2^a + 2^b$) is the simplest so far, but it becomes extremely large for quite small values of $a$ and $b$. For a more manageable function, I suggest interleaving the binary representations of $a$ and $b$. Then the result will be no larger than $\max(a,b)^2$.

To make it commutative ($f(a,b)=f(b,a)$), you will first have to swap $a$ and $b$ if $a< b$. Then it goes something like this:

f := 1
while a != 0 or b != 0
    // Incorporate bottom bit of a
    f := 2 * f
    if a is odd then f := f + 1
    a := a/2 // Discard bottom bit of a

    // Incorporate bottom bit of b
    f := 2 * f
    if b is odd then f := f + 1
    b := b/2 // Discard bottom bit of b
wend

Answer 5

Note: This answer is substantially the same as the one given by JimmyK4542. I am leaving it here in case some minor difference in wording helps someone understand the derivation.

If we can additionally assume that the integers are nonnegative, I believe that the following will satisfy the conditions given.

First note that commutativity can be guaranteed by sorting the elements, so without loss of generality we assume $a \geq b$. Call the given pair which has been sorted $(a_0, b_0)$. We can identify the following sequence which uniquely transforms a sorted pair $(a,b)$:

\begin{aligned} (0,0) &\rightarrow 0 \\ (1,0) &\rightarrow 1 \\ (1,1) &\rightarrow 2 \\ (2,0) &\rightarrow 3 \\ (2,1) &\rightarrow 4 \\ (2,2) &\rightarrow 5 \\ &\vdots \end{aligned}

From this it is clear that if we can compute the number of elements in this sequence which have $a < a_0$, and then add $b_0$, we have an answer that works. Let \begin{equation} N(k) = \sum_{n=0}^{k}{n} = (k)(k+1)/2 \end{equation}

Then we have a mapping \begin{equation} (a \, \& \, b) \rightarrow N(max(a,b))+min(a,b) \end{equation} which satisfies the two properties given.

As a benefit, this answer also scales only quadratically with the largest number in the pair.

Answer 6

More programmerly than mathily, use a nondigit separator, like the convenient decimal point. In this case I concat a string together then cast it to a float.

c = parseFloat(max(a,b) + '.' + min(a,b))

c will be unique and reversible for all interchangeable combinations of a and b.

so for example,

myhash(124,24) = 124.24
myhash(24,124) = 124.24

myhash(11231,26611) = 26611.11231

I think some index systems like the Dewey Decimal system and some part numbering schemes use this sort of bin-based technique.

But oops, then there is a problem if the second number ends with trailing zeros, so stringwise reverse it to preserve them:

c = parseFloat(max(a,b) + '.' + reverse(min(a,b)))

then

myhash(123,456) = 123.654
myhash(12300,4560) = 12300.0654

Answer 7

Sort a and b ,then apply hash function.

c=hash(sort(a,b))

Here,

1. c is unique for the inputs and not reversible.
2. a and b are interchangeable (a & b = b & a).

Answer 8

Let $p$ and $q$ be distinct primes, and let

$$c=\min(p^aq^b,p^bq^a)$$

In fact this generalizes to a function for $n$ interchangeable variables $a_1,\ldots,a_n$, using distinct primes $p_1\ldots,p_n$:

$$c=\min_{\pi}\{\prod_{i=1}^np_i^{a_{\pi(i)}}\}$$

where the min is taken over all permutations of $\{1,\ldots,n\}$.

(Note, the integers $a$ and $b$ need not be positive. The Fundamental Theorem of Arithmetic still guarantees that different multisets correspond to different values of $c$.)

Answer 9

Well, there is a rather simple way to do this using string operations:

1. Let x = min(a,b) and y=max(a,b)
2. Treating x and y as strings and + as the concatenation operator, let z = x + "|" + y
3. Use your favorite string to binary/hex/decimal function to turn z into a number.

Let's look at an example. f(25,-36).

x = -36 and y = 25.

z = "-36|25".

Using a standard ASCII to Decimal converter, we get the number 4551541245053.

Answer 10

Given $\max(|a|,|b|)$ and $a+b$, we can recover the unordered pair $(a,b)$ as $$ \textstyle\left(\min(a+b,0)+\max(|a|,|b|),\max(a+b,0)-\max(|a|,|b|)\right)\tag{1} $$ There are $4n+1$ unordered pairs of integers so that $\max(|a|,|b|)=n$; their sums being $$ \{-2n,-2n+1,\dots,0,\dots2n-1,2n\}\tag{2} $$ Since $$ \sum_{k=1}^n(4k+1)=2n^2+3n\tag{3} $$ we will set $$ f(a,b)=2\max(|a|,|b|)^2+\max(|a|,|b|)+a+b\tag{4} $$ Then the greatest $f(a,b)$ can be for $\max(|a|,|b|)=n$ is $$ 2n^2+n+2n=2n^2+3n\tag{5} $$ and the least it can be for $\max(|a|,|b|)=n+1$ is $$ 2(n+1)^2+(n+1)-2(n+1)=2n^2+3n+1\tag{6} $$ Therefore, the $f$ given in $(4)$ maps unordered pairs of integers to non-negative integers injectively (and incidentally, surjectively).

Answer 11

If $P$ are all the primes, then:

$f(a,b) = P(a) P(b)$

f(0,0)=4
f(1,0)=6
f(1,1)=9
f(2,0)=10
f(2,1)=15
f(2,2)=25
f(3,0)=14
f(3,1)=21
f(3,2)=35
f(3,3)=49

$a$ & $b$ are interchangeable. It is reversible, but very hard for large numbers of c. I think this is also how asymmetric cryptography works in its most simple form with large numbers of '$a$' & '$b$'.

Example:

• 'P(a)' being the public-key on the server side
• 'P(b)' being the private-key on the client side

• 'c' being the token to be transmitted.

  1. client request access to server
  2. server transmits token to client and requests for a factor of it.
  3. client divides token with the locally stored private-key and send the result to the server
  4. Server compares stored public-key with given result, if same then grants access

Answer 12

Since your answer need to be a integer the following simple steps should work:

1. Sort the numbers a and b with the smallest first if a = b just put a first.
2. Now put the first number in front of the dot and the second number behind the decimal, then append one to the end (so that if the second number ended in 0 it will now end in 01 to keep it unique)
3. now you have a unique easily reversible number.

This method will also be reasonably memory efficient, and also ensure uniqueness (and you can easily find a given b and b given a).

Answer 13

Use complex numbers $u+vI$ with integer coefficients $u, v$. Let $c=a*b=(a+b)+abI$. It is easy to see that if we know $u=a+b$ and $v=ab$ then we know $a, b$. To see this consider the polynomial $x^2-ux+v$ then this has roots $a,b$.

If you want the answer to be real then you could use $(a+b)+(ab)\pi$

Answer 14

Yet another (and quite standard) solution is using the Cantor pairing function $\pi: \mathbb N\times\mathbb N\to \mathbb N$. It is a bijection. So now let's only find a bijection between unordered pairs ${\mathbb N}\choose{2}$ and ordered pairs $\mathbb N\times \mathbb N$. One such is $$\sigma:\{a,b\}\mapsto(\min\{a,b\},\max\{a,b\}-\min\{a,b\})$$ with the inverse $$\sigma^{-1}:(x,y)\mapsto \{x,x+y\}.$$

Therefore $\pi\circ\sigma$ satisfies your requirements, moreover, it is a bijection between unordered pairs of natural numbers and natural numbers, so it's efficient.

If you need integers instead of natural numbers, use your favourite bijection $\mathbb Z\to\mathbb N$, there's plenty of them.

Answer 15

Since some seem to be talking about programming and this is too long for a comment, I'll post this answer even though it's not really appropriate for a mathematics question. Given two 16 bit signed ints a and b, you can sort them and or them together. Meaning:

int uniqueCombo(int a, int b) { //assuming -2^15<=a,b<=2^15-1
    if(a<b)
        return (a&0xFFFF)|(b<<16);
    else
        return (b&0xFFFF)|(a<<16);
}

This, by the magic of signed and unsigned integers, unless I'm mistaken, will handle positives and negatives perfectly fine and is easy to invert (to get a as a 32 bit signed int, take out the first sixteen bits. If the biggest place digit is a one, pad the new 16 large place digits with ones. Else zeros.)