# Are All Perfect Numbers Even?

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## Open Question

By the Theorem of Even Perfect Numbers, it is known that an even number is perfect if and only if it is of the form:

- $2^{n - 1} \paren {2^n - 1}$

where $2^n - 1$ is prime.

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

## Progress

### Minimum Size of Odd Perfect Number

It had been established by $1986$ that an odd perfect number, if one were to exist, would have over $200$ digits.

By $1997$ that lower bound had been raised to $300$ digits.

By $2012$ that lower bound had been raised again to $1500$ digits.

### Form of Odd Perfect Number

An odd perfect number $n$ is of the form:

- $n = p^a q^b r^c \cdots$

where:

- $p, q, r, \ldots$ are prime numbers of the form $4 k + 1$ for some $k \in \Z_{>0}$
- $a$ is also of the form $4 k + 1$ for some $k \in \Z_{>0}$
- $b, c, \ldots$ are all even.

### Prime Factors of Odd Perfect Number

An odd perfect number has:

- at least $8$ distinct prime factors
- at least $11$ distinct prime factors if $3$ is not one of them
- at least $101$ prime factors (not necessarily distinct)
- its greatest prime factor is greater than $1 \, 000 \, 000$
- its second largest prime factor is greater than $1000$
- at least one of the prime powers factoring it is greater than $10^{62}$
- if less than $10^{9118}$ then it is divisible by the $6$th power of some prime.

## Sources

- 1919: Leonard Eugene Dickson:
*History of the Theory of Numbers: Volume $\text { I }$*... (previous) ... (next): Preface - 1971: George E. Andrews:
*Number Theory*... (previous) ... (next): $\text {3-5}$ The Use of Computers in Number Theory: Conjecture $2$ - 1986: David Wells:
*Curious and Interesting Numbers*... (previous) ... (next): $2$ - 1986: David Wells:
*Curious and Interesting Numbers*... (previous) ... (next): $28$ - 1989: Ephraim J. Borowski and Jonathan M. Borwein:
*Dictionary of Mathematics*... (previous) ... (next): Entry:**perfect number** - 1992: George F. Simmons:
*Calculus Gems*... (previous) ... (next): Chapter $\text {B}.2$: More about Numbers: Irrationals, Perfect Numbers and Mersenne Primes - 1994: Richard K. Guy:
*Unsolved Problems in Number Theory*(2nd ed.) - 1997: David Wells:
*Curious and Interesting Numbers*(2nd ed.) ... (previous) ... (next): $2$ - 1997: David Wells:
*Curious and Interesting Numbers*(2nd ed.) ... (previous) ... (next): $28$

- Greathouse, Charles and Weisstein, Eric W. "Odd Perfect Number." From
*MathWorld*--A Wolfram Web Resource. https://mathworld.wolfram.com/OddPerfectNumber.html