January 19, 2015

Horse 1825 - The No Odd Perfect Numbers Conjecture

In Horse 1680 I laid out the search for the first boring number and found that every number up to 75 is interesting in some way before  having a metaphorical brain explosion and crashing an imaginary¹ car into an imaginary telephone pole.
During that search I said that both 6 and 28 were interesting because they are Perfect numbers.

A Perfect number is one whose proper positive divisors sum together to make the number.
6 is a perfect number because its proper positive divisors of 1, 2, & 3 add together to make 6. 1+2+3=6.
28 is a perfect number because 1+2+4+7+14=28.
The next two perfect numbers are 496 and 8128; you might like to check that out in your spare time.

Currently it is unknown whether there are any odd perfect numbers as none have ever been discovered. I suspect though (and I'm not a mathematician by any stretch of the imagination²) that there are no perfect odd numbers.

Observation 1:
All odd numbers only have other odd numbers as their proper positive divisors.
Odd x Odd = Odd
Odd x Even = Even
Even x Even = Even
Any even number in a factorisation, instantly produces an even number.
For instance, 75's factors are 1, 3, 5, 15, 25 and 75. All are odd.

Observation 2:
There must be an odd number of odd divisors.
Odd + Odd = Even
Odd + Even = Odd
Even + Even = Even
Odd + Odd + Odd = Odd etc.
Take the number 15. 1+3+5=9 This is deficient but still odd. There can not be an even number of divisors since that produces an even number. By definition, a perfect odd number is odd,

Observation 3:
Perfect numbers follow the same sorts of criteria as abundant number.
A number is said to be abundant, if the sum of the factors exceeds the original number. 12 is abundant as 12's factors of 1 + 2 + 3 + 4 + 6 = 16 and 16 >12.
The smallest odd abundant number³ is 945. 1+3+5+7+9+15+21+27+35+45+63+105+135+189+315= 975 and 975 > 945. (945 has 15 divisors)
The fact that there are abundant odd numbers rules in the possibility that perfect odd numbers could exist. Whether they do or not is another question.

Observation 4:
Abundant numbers are not semi-primes. A semi prime has two prime factors that are not one.
When you multiply two numbers together you can not end up with a number with more significant figures than the total number of significant figures that the two numbers you multiplied together had; since a semi-prime only has two factors other than one, then that's simply not enough factors to add together to get close to the final answer.
Take a small semi prime number like 77. 77 = 7x11. 1+7+11 = 19 which is scandalously deficient. This only gets worse with bigger numbers.
If abundant numbers are not semi-primes, then odd perfect numbers aren't likely to be as the first abundant odd number is 945.

Observation 5:
All currently observed perfect numbers are of the for PN= (2n-1)(2n-1)
What's of note there is that 2n for any n including n-1 which is just another n, is always going to be an even number because 2 is even. Since (2n-1) is always going to be an even term, the whole expression which generates all of the currently known perfect number can and only must generate even numbers because Odd x Even = Even

Observation 6:
If odd perfect numbers aren't semi-primes, then they must be multi-composite.
This should be obvious to all as even the first abundant number of 12 has three ways to get there 1x12, 2x6 and 3x4.
The reason I make mention this is that apart from 6 which is small, the next three perfect numbers are many times composite.
28 has three ways to get there, 1x28, 2x14 and 4x7
496 has five ways to get there, 1x496, 2x248, 4x124, 8x62 and 16x31.
8128 has seven ways to get there, 1x8128, 2x4064, 4x2032, 8x1016, 16x508, 32x254 and 64x127.
I think that it follows that if the first perfect odd number is greater than the first abundant odd number, we know that this must be true since there are no perfect odd numbers less than 945, then just like the known perfect even numbers, the first perfect odd number must be many times composite.

Now I know that the first 945 numbers out of infinity is a pathetically small sample size but I do not possess the mathematical know-how to prove the case for all numbers. I do know through computational blunt force all numbers to 10^300 have been tested, which is a number so big that it looks like this:

The thing is that the first perfect odd number if it even exists, is bigger than this and must be so hideously mutil-composite and yet fulfil all of the other conditions that I've mentioned, that it just seems unlikely. At any rate, such a number if it does exist, is so humongous as to be totally pointless.
I don't think that there are any perfect odd numbers but that still hasn't been proven by anyone. I just think that there are so many conditions required to produce one, and the fact that no do exist up to 10^300, that none will exist. Again, it hasn't been proven and that's why it's only a conjecture.

¹An imaginary car is one which is the square root of a negative car. Try not to think about that much. It's a bit complex.
²Does the square root of anti-matter equal imaginary matter? If so, if we take our imaginary car and square it, would it create an anti-matter car? If such a car were to be involved in a road accident with its real matter counterpart, both would spontaneously disappear in a flash of energy - that would make for an interesting episode of Crash Investigation Unit on Channel 7, wouldn't it?
³I only found this through brute force by testing every single odd number until I found one. It took 57 minutes.

1 comment:

Anonymous said...

The number you have printed is 10^720, not 10^300.

The following maple program lists every odd abudant integer up to 10^4 in less than a minute.
for i from 1 to 10000 by 2 do s := 0: for d from 1 to i/3 do if i mod d = 0 then s := s+d: fi: od: if s > i then print(i, s) fi: od:

The wikipedia page about perfect numbers is much more informative and interesting than this post, so why bother ?