# Are All Triperfect Numbers Even?

Jump to navigation
Jump to search

## Contents

## Open Question

The sequence of triperfect numbers begins:

- $120, \quad 672, \quad 523 \, 776, \quad 459 \, 818 \, 240, \quad 1 \, 476 \, 304 \, 896, \quad 51 \, 001 \, 180 \, 160$

These are all the triperfect numbers that are currently known.

As can be seen, they are all even.

It is not known whether there exist any odd triperfect numbers. None have ever been found.

## Progress

### Minimum Size of Odd Triperfect Number

It has been established that an odd triperfect number, if one were to exist, would be greater than $10^{70}$.

If it does not have $3$ as a prime factor, then it is greater than $10^{108}$.

### Form of Odd Triperfect Number

An odd triperfect number is square.

### Prime Factors of Odd Triperfect Number

An odd triperfect number has:

- at least $11$ distinct prime factors
- at least $32$ distinct prime factors if $3$ is not one of them.

## Sources

- 1986: David Wells:
*Curious and Interesting Numbers*... (previous) ... (next): $120$ - 1997: David Wells:
*Curious and Interesting Numbers*(2nd ed.) ... (previous) ... (next): $120$