Download Determining if any Odd Perfect Numbers Exist.

Determining if any Odd
Perfect Numbers Exist.
CORYDEN ENOCHS
“The world was created in 6 days and the moon orbits in 28 days because 6 and 28 are perfect.” –Philo of Alexandria
What is a perfect number?
A number N is perfect if it is equal to twice the sum of its divisors: 𝜎 𝑁 = 2𝑁.
For example:
6 is the first perfect number. Its positive divisors are 1, 2, 3, and 6. So, 𝜎 𝑁 = 12 and
2𝑁 = 12.
The second perfect number is 28. 𝜎 𝑁 = 1 + 2 + 4 + 7 + 14 + 28 = 56 and 2𝑁 = 56.
The list of perfect numbers continues with 496,8128,33550336,8589869056,...
Another view of perfect numbers is to recognize that the sum of the positive proper divisors is
the original number.
Some History
The definition of perfect numbers was first found in Euclid’s Elements. The first four perfect
numbers are considered to have been discovered by Euclid. The fifth was seen referenced in a
manuscript written by an unknown mathematician between 1456-1461. Pietro Cataldi showed
the sixth and seventh perfect numbers in 1588.
Even Perfect Numbers
As we have seen, the list of perfect numbers contains only even numbers so far. It was proved by
Euclid that 2𝑝−1 2𝑝 − 1 is an even perfect number whenever (2𝑝 − 1) is prime.
We can see the first four generated in this form with p=2,3,5,7
p=2: 21 22 − 1 = 2 4 − 1 = 2 ∗ 3 = 6
p=3: 22 23 − 1 = 4 8 − 1 = 4 ∗ 7 = 28
p=5: 24 25 − 1 = 16 32 − 1 = 16 ∗ 31 = 496
p=7: 26 27 − 1 = 64 128 − 1 = 64 ∗ 127 = 8128
Prime Numbers of the form 2𝑝 − 1 are known as Mersenne Primes.
In the 18th century, Euler proved that this formula will yield ALL even perfect numbers. This is
known as the Euclid-Euler Theorem.
How Many are There?
There is a one-to-one relationship between Mersenne Primes and perfect numbers, so each
Mersenne Prime generates one perfect number. As of April 2015, 48 Mersenne Primes have
been discovered, so there are currently 48 known perfect numbers. It is not yet known if there
are infinitely many perfect numbers, or if there are infinitely many Mersenne Primes. In other
words, there could potentially be a ceiling, after which no other perfect numbers exist. But I will
leave this to another discussion.
Odd Perfect Numbers
While there are no known odd perfect numbers, and it has been conjectured that none even
exist, there have been a variety of results regarding necessary conditions that must hold true of
a perfect odd number.
One such result is a lower bound for possible solutions. This is currently 101500 . This is simply an
immensely large number as a lower bound for any possible odd perfect numbers.
Another result concerns the number of distinct prime factors. This has currently been proven to
require at least 10 distinct prime factors. If not considering distinctness, the number of prime
factors has gone up as high as 101 prime factors. It has also been shown that the greatest prime
factor must be greater than 108 , the second largest greater than 104 , and the third largest
greater than 100.
1500
10
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