Odd Perfect numbers and using science proof

[Bertie Wooster]

Perfect numbers are numbers whose whole-number factors, when added, equal the number itself. For example, 6 is a perfect number because it's factors: 1,2,3 all added together equals 6. 28 is another perfect number since it's factors; 1,2,4,7, 14 add up to 28.

All the perfect numbers that we know are even numbers. However, in terms of mathematics(number theory), you cannot prove that no perfect number is an odd number.

This seems like bullshit to me. Just because we can't use an equation to prove it means it can't be proven? How about using scientific principles (or logic) to prove certainty that there are no odd perfect numbers? Can't logic be used in numbers theory?

[The Grim Squeaker]

There is a difference between mathematical proof (Nothing will contradict) and logical proof (We know of nothing that can contradict) I may have gotten mixed up here with logical proof though.

[Darth Wong]

Science does not prove; science concludes, based on evidence.

Mathematics can "prove" something because it defines its own rules. All mathematical proofs are proof through "definition + logical deduction".

[RedImperator]

Math and science aren't the same things. Using scientific reasoning, you can conclude there are probably no odd perfect numbers, which in the real world is as good as saying there aren't any, but in mathmatics, that's not good enough.

[Vicious]

Based on the facts, an odd perfect number is impossible. Any whole number multiplyed by two will always result in an even number. Since a perfect number is simply a number who's factors add together to equal itself, it can be rewritten as a multiplication problem. I'll use the number 28 to demonstrate.

Factors of 28: 1, 2, 4, 7 and 14.

14 = 1/2 of 28.
1 + 2 + 4 + 7 = 14 = 1/2 of 28.

Therefore, 1 + 2 + 4 + 7 + 14 = 14 + 14 = 14 X 2 = 28.
I think you'll find the same to hold true for any perfect number. Since the highest factor a whole number N can have is 1/2N, the other factors can never add up to more than 1/2N.

Where in number theory does it state that it can't be proven there is no odd perfect number? Logic dictates that it is in fact impossible. Since the above statements are true, you can conclude that a perfect number is impossible.

Perhaps by mathematical standards, without calculating every single perfect number, you can't state definitively whether the above is true. However, I think that mathematics must also bow to logic in cases such as this. It is theoretically impossible to calculate every single perfect number. However, if you can calculate a sufficient number of them and find that the above holds true, then you can say that the above is true.

[Kuroneko]

All the perfect numbers that we know are even numbers. However, in terms of mathematics(number theory), you cannot prove that no perfect number is an odd number.

No. In mathematics, there is no theorem stating the impossibility of proving that no perfect numbers are odd. There is, however, a lack of proof of either possibility or impossibility. That kind of distinction might, at first glance, appear to be overly pedantic, but it is a vastly different situation for a mathematician. Whether or not there are odd perfect numbers is a relatively famous unsolved problem in number theory.

Based on the facts, an odd perfect number is impossible. Any whole number multiplyed by two will always result in an even number. Since a perfect number is simply a number who's factors add together to equal itself, it can be rewritten as a multiplication problem. I'll use the number 28 to demonstrate.
Factors of 28: 1, 2, 4, 7 and 14. 14 = 1/2 of 28. 1 + 2 + 4 + 7 = 14 = 1/2 of 28. Therefore, 1 + 2 + 4 + 7 + 14 = 14 + 14 = 14 X 2 = 28.
I think you'll find the same to hold true for any perfect number. Since the highest factor a whole number N can have is 1/2N, the other factors can never add up to more than 1/2N.

Incorrect. Constructing a counterexample is pathethetically easy: 5!! = 15 < 2*[1+3+5] = 2*9. Additionally, if you think that the sum of the proper factors of a number must be strictly less than or equal to the original number, that is false as well: try 4! = 24, or 9!! = 945 if you prefer odd numbers.

Where in number theory does it state that it can't be proven there is no odd perfect number?

Nowhere. See my reply to the OP.

However, I think that mathematics must also bow to logic in cases such as this.

Which logic?

[CaptainChewbacca]

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434790271942656

Enjoy!

[Kuroneko]

Let N =

931144559095633232126208229355238422113850001463928526...434790271942656
Enjoy!

Then N = 2^{p-1}[2^p-1], where p = 19937, which is a prime number. The Lucas-Lehmer test confirms the primehood of 2^p-1. QED.

[CaptainChewbacca]

Its the largest known perfect number.

[Kuroneko]

Its the largest known perfect number.

As far as I am able to determine, the largest known perfect number as of February of this year has 15,632,458 digits [1]. It is by far too big to post on a message board.

[RedImperator]

Its the largest known perfect number.

As far as I am able to determine, the largest known perfect number as of February of this year has 15,632,458 digits [1]. It is by far too big to post on a message board.

Well, maybe not too big to post technically, but definitely too large to post if you care to remain a member of that message board.

[Vicious]

Based on the facts, an odd perfect number is impossible. Any whole number multiplyed by two will always result in an even number. Since a perfect number is simply a number who's factors add together to equal itself, it can be rewritten as a multiplication problem. I'll use the number 28 to demonstrate.
Factors of 28: 1, 2, 4, 7 and 14. 14 = 1/2 of 28. 1 + 2 + 4 + 7 = 14 = 1/2 of 28. Therefore, 1 + 2 + 4 + 7 + 14 = 14 + 14 = 14 X 2 = 28.
I think you'll find the same to hold true for any perfect number. Since the highest factor a whole number N can have is 1/2N, the other factors can never add up to more than 1/2N.

Incorrect. Constructing a counterexample is pathethetically easy: 5!! = 15 < 2*[1+3+5] = 2*9. Additionally, if you think that the sum of the proper factors of a number must be strictly less than or equal to the original number, that is false as well: try 4! = 24, or 9!! = 945 if you prefer odd numbers.

Alright, I feel like I'm off the deep end without a lifevest, but I can see at least one mistake I made. What I meant, and forgot to type as I thought, was "Since the highest factor an even whole number N can have is 1/2N, the other factors can never add up to more than 1/2N."

As to the points, I shall concede them, since my knowledge of mathematics is limited and I apparently was unaware of facts which render my idea incorrect.

[Kuroneko]

Alright, I feel like I'm off the deep end without a lifevest, but I can see at least one mistake I made. What I meant, and forgot to type as I thought, was "Since the highest factor an even whole number N can have is 1/2N, the other factors can never add up to more than 1/2N."

This is still dead wrong, for much the same reasons, except that now the 4! counterexample does not work (all the rest do, however). It is quite trivial to verify that the factors of 15 are 1, 3, and 5, and 1+3+5 = 9 > 15/2. For an even counterexample, 12 is nice: 1+2+3+4+6 = 16, which is greater than any of (1/2)12, 1/(2*12), and 12 itself. An odd counterexample that has a sum of factors greater than the original number (not just half the original), try 9!! = 945.

[Vicious]

Alright, I feel like I'm off the deep end without a lifevest, but I can see at least one mistake I made. What I meant, and forgot to type as I thought, was "Since the highest factor an even whole number N can have is 1/2N, the other factors can never add up to more than 1/2N."

This is still dead wrong, for much the same reasons, except that now the 4! counterexample does not work (all the rest do, however). It is quite trivial to verify that the factors of 15 are 1, 3, and 5, and 1+3+5 = 9 > 15/2. For an even counterexample, 12 is nice: 1+2+3+4+6 = 16, which is greater than any of (1/2)12, 1/(2*12), and 12 itself. An odd counterexample that has a sum of factors greater than the original number (not just half the original), try 9!! = 945.

As I said, I concede the points. My knowledge of mathematics fell short of what I needed, so I made those statements out of ignorance. I merely wished to fix the typographical error.

[Dooey Jo]

Actually, I think there's a big misunderstanding among much of the population about what logic actually is. In the OP it is suggested that we should perhaps use logic to prove that there are no odd perfect numbers. But that is exactly what mathematicians have tried to do for all these years, because all mathematical proofs must come through logical deduction (just like Wong said).

I think many (if not most) believe that "logical" simply means "reasonable". It would of course seem reasonable that there are no odd perfect numbers, but that does not mean that it's logical to state there aren't any (unless there's a system of logic that I've never heard of, that states that "reasonable therefore true" is valid).

My theory is that this is Star Trek's fault. With Spock and Tuvok running around claiming this and that is or is not logical, even though there are usually no way of applying real logic to that situation. Or maybe ST simply perpetuated an already widespread misconception...

I wonder, is there a fallacy called "Appeal to common sense"?

[Rye]

Perfect numbers are numbers whose whole-number factors, when added, equal the number itself. For example, 6 is a perfect number because it's factors: 1,2,3 all added together equals 6. 28 is another perfect number since it's factors; 1,2,4,7, 14 add up to 28.

All the perfect numbers that we know are even numbers. However, in terms of mathematics(number theory), you cannot prove that no perfect number is an odd number.

This seems like bullshit to me. Just because we can't use an equation to prove it means it can't be proven? How about using scientific principles (or logic) to prove certainty that there are no odd perfect numbers? Can't logic be used in numbers theory?

You're equivocating on the term "prove" and "proof"..one is the mathmatical term, and one is the vernacular.

But uh, am I missing something here? Don't all perfect numbers have a half of total value in their factors? So doubling that will always result in an even number.

[Rye]

One USE is the mathmatical term, not one word (prove & proof).**Just ot make that explicitly clear.

[Itô Doeblin]

But uh, am I missing something here? Don't all perfect numbers have a half of total value in their factors? So doubling that will always result in an even number.

All even numbers have a half of total value in their factors. But as Kuroneko wrote, the question of whether all perfect numbers are even is still open. And should there be an odd perfect number, this perfect number would obviously not have 2 as a factor, and thus also not half of their total value.

Have a nice day, Itô

[Rye]

But uh, am I missing something here? Don't all perfect numbers have a half of total value in their factors? So doubling that will always result in an even number.

All even numbers have a half of total value in their factors. But as Kuroneko wrote, the question of whether all perfect numbers are even is still open. And should there be an odd perfect number, this perfect number would obviously not have 2 as a factor, and thus also not half of their total value.

Yeah, I realised that when I was playing Battlefield 2 before, it just didn't occur to me when I posted. I saw the other examples and didn't realise that it was coincidental, rather than a specific part of what makes a perfect number.

[TimothyC]

Wouldn't such a number (an odd perfect number) have to not be a square of another odd number to get around having an even number of factors to add up? And wouldn't that then make the number of possible odd perfects a little bit smaller, and thus easier to search? Or am I being crazy (a very good possibility right now)?

[Kuroneko]

Wouldn't such a number (an odd perfect number) have to not be a square of another odd number to get around having an even number of factors to add up? And wouldn't that then make the number of possible odd perfects a little bit smaller, and thus easier to search? Or am I being crazy (a very good possibility right now)?

You're not crazy (or, at least, there is no evidence of any craziness here); that's a good observation. If N = Prod_k[p_k^{a_k}], where p_k are primes, e.g., 36 = [2^2][3^2], then any given divisor d can include up to a_k factors of p_k, making the total number of divisors is simply Prod_k[a_k+1], e.g., for N = 72, the total number of divisors is (2+1)(2+1) = 9. If N is odd and is a square of another number, then all of a_k are even, and hence the total number of divisors is odd, making the number of divisors strictly less than N even (since N itself has to be excluded), so the sum of them must be even (being the sum of an even number of odds).

However, that's not really that good of a test for the [im]possibility of a specific odd number being perfect. Asymptotically, the probability of a random integer being a square is zero, so for large numbers, chances are that the test will simply say that the given odd number is not a square, which is an inconclusive result either way. On the other hand, it is known that any odd perfect number must have a prime factor greater than 10^6, meaning this comparatively easy test alone gets rid of almost 96% of the candidates. Additionally, there are other criteria, such as are bounds on the number of prime factors for odd perfect numbers, etc. It is known that there is no odd number less than 10^300 is perfect.

[Enola Straight]

AFAIK, not only are there no odd perfect numbers, but no perfect numbers which do not end in the digits 6 or 8.

[Zero]

AFAIK, not only are there no odd perfect numbers, but no perfect numbers which do not end in the digits 6 or 8.

What about the one in Chewbacca's post? It ends in 6. Is that number not a real perfect number?

[StimNeuro]

AFAIK, not only are there no odd perfect numbers, but no perfect numbers which do not end in the digits 6 or 8.

What about the one in Chewbacca's post? It ends in 6. Is that number not a real perfect number?

It is. He said there does not exist a perfect number that does not end in 6 or 8.