Perfectly odd

Submitted by Marianne on February 5, 2020

Do odd perfect numbers exist?

A perfect number is a whole number which equals the sum of its proper divisors: for example, $6$ is divisible by $2,3$ and $1$ and is also equal to the sum $1+2+3=6.$ Similarly, $28$ is divisible by $1,2,4,7$ and $14$ and equal to $1+2+4+7+14=28.$

Perfect numbers are easy to define, but finding examples of them is a different matter: although mathematicians have been looking for them since the time of the ancient Greeks, they have only managed to find 51 so far (you can find out more about the history of perfect numbers here).

One thing all the known perfect numbers have in common is that they are even. But what about odd perfect numbers? Do they exist, and if yes, what are they?

What would odd perfect numbers look like?

In the eighteenth century the legendary mathematician Leonhard Euler showed that an odd perfect number $n,$ if it exists, must look like

$n=p^ j(m^2),$

where the numbers $p$ , $j$ and $m$ satisfy the following restrictions:

The number $m$ is an odd integer,
The number $p$ is a prime number and of the form $4k+1$ for some positive integer $k$ ,
The number $j$ is also of the form $4k+1$ for some positive integer $k$ (though it doesn't need to be prime).

Let's illustrate this with an example. The first prime number of the form

$p=4k+1$

$5= 4\times 1 +1.$

For the exponent $j$ we can choose

$j=5=1\times 4+1$

to get

$p^ j=5^5=3125.$

Choosing $m=3$ for the odd number $m$ gives us

$n=5^5 3^2=28125$

as a candidate for an odd perfect number.

We can see that even if we choose small values for $p,j$ and $m$ , the resulting candidate $n$ is already quite large. Indeed, Euler’s result means that there are only eleven odd perfect number candidates under $100$ , namely

$5,13,17,33,37,41,45,53,73,89,\; \; \mbox{and}\; \; 97.$

More restrictions

We can whittle this list down further using a result proved in 1953, two full centuries after Euler, by the French mathematician Jacques Touchard. He showed that every odd perfect number $n$ must also be of the form

$n=12k + 1 \; \; \; \mbox{or} \; \; \; n=36k + 9,$

where $k$ as before is a positive integer.

In our list of eleven numbers above, this only applies to four, namely

$\displaystyle 13$	$\displaystyle =$	$\displaystyle 1 \times 12+1$
$\displaystyle 37$	$\displaystyle =$	$\displaystyle 3 \times 12 +1$
$\displaystyle 45$	$\displaystyle =$	$\displaystyle 1 \times 36+9$
$\displaystyle 97$	$\displaystyle =$	$\displaystyle 8\times 12+1$

Is any of these four numbers perfect? You can check for yourself that the answer is no. We have just proved that there are no odd perfect numbers under $100.$ Over the centuries mathematicians have come up with a number of other restrictions on the form any odd perfect numbers must take (you can see a list here). These don't help us prove that odd perfect numbers do or don't exist, but at least they help us to rule out candidates.

How big are odd perfect numbers?

Calculations like the ones above can be done with just your brain, paper and a pencil, but once you have a computer at your disposal you can go much, much further in eliminating odd perfect number candidates. This is just what mathematicians have done, using mathematical results combined with clever computer algorithms. The table below illustrates the advances that have been made over the last few decades.

Year	There's no odd perfect number under
1957	$10^{20}$
1973	$10^{50}$
1989	$10^{160}$
1991	$10^{300}$
2012	$10^{1500}$

Note that the last result means that any odd perfect number, if it exists, must have at least $1500$ digits. That's absolutely huge and has led mathematicians to suspect that odd perfect numbers probably don't exist at all. But a suspicion isn't a proof, no matter how much evidence counts in its favour. Until mathematicians develop the tools required for a proof, the mystery of odd perfect numbers remains open.

About the author

Dylan Johnson is a college student who will be starting a doctorate program in maths next fall. He is particularly fond of number theory, because he adores how some of its most accessible problem statements — such as perfect numbers, Fermat's Last Theorem, and the distribution of prime numbers — require intricate and complex solutions. When he's not thinking about number theory, he likea rock climbing, listening to music, and playing games.