The Six Triperfect Numbers - Numberphile
video description
Assume k is an odd triperfect number. Then the sum of its factors must exactly equal 3k, which is itself an odd number. Now, in grade school we learned a technique to calculate the sum of the factors of k, which went: -
Break k into its prime factors: k = p1-n1 - p2-n2 - p3-n3 -. - pm-nm, where p1, p2, p3, , pm are the prime factors of k. Then the sum of the factors of k equals: -
(1 + p1 + p1-2 +. + p1-n1) (1 + p2 + p2-2 +. + p2-n2) (1 + p3 + p3-2 +. + p3-n3). (1 + pm + pm-2 +. + pm-nm) = 3k-
Since 3k is odd, all of the factors in the above formula must be odd, which is possible only if every exponent in the prime factorization is even. Therefore, k must be a perfect square. Now, if anyone can prove a triperfect number cannot be a perfect square, we have proven there are no odd triperfect numbers.
- - - - - -
We can use similar reasoning to at least figure out some charactistics of an odd perfect number, if one exists. We know the sum of its factors must exactly equal 2k. Since k is odd, we know 2k does NOT divide 4. That means in the prime factorization of k, exactly one of the exponents must be odd and all others must be even. It follows that k must have the form:
p - n-2
In other words, an odd perfect number must be the product of a prime number p and an odd perfect square.
But wait, we can do better: p is the prime factor that has an odd exponent in the factorization of k. If p is 3 more than a multiple of 4, then (1 + p + p-2 +. + p-n) in the factorization of k must be a multiple of 4, contradicting the fact that 2k cannot be a multiple of 4. (This is because p-n = 3 mod 4 when n is odd, p-n = 1 mod 4 when n is even) So, p cannot be 3 more than a multiple of 4.
We now know that p must be one more than a multiple of 4. But if that is the case, then the exponent n associated with p in the factorization of k cannot be 3 more than a multiple of 4 because (1 + p + p-2 +. + p-n) will be a multiple of 4, once again contradicting the fact that 2k is not a multiple of 4. In conclusion, an odd perfect number must be of the form p-n - m-2, where
(a) p = 1 mod 4 and prime
(b) n = 1 mod 4
(c) m does not have 2 or p as a factor
Date: 2022-04-09
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Comments and reviews: 9
Zahid
Answering the question about -odd- perfect number I have a -conjecture- and you can always check it and let me know if I am correct.
My conjecture is that all the perfect numbers are constructed by the formula given below or at least the reverse is true i. e. if any number which fits this formula has got to be a perfect number.
2-n x (2-(n+1)-1)
Provided that the number inside the outer set of bracket is a prime number.
I will explain it by example.
If you put 1, 2, and 4 in place of n, you get the first three perfect numbers i. e. 6, 28, and 496.
If the n is 3 the number inside the bracket becomes 15 which is not a prime number so 120 is not a perfect number though incidentally ) it is the first tri perfect number.
Having said that I 'intuitively' know that second part of my conjecture is correct. Or maybe it is not just intuition but something to do with the fact that the factors of the 2-n are 2-(0-n) and nothing else. I would very much welcome if it is contrary to that.
Now if the first part of my conjecture is also correct then all the perfect numbers have to be even. I would very much appreciate the input on this.
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Answering the question about -odd- perfect number I have a -conjecture- and you can always check it and let me know if I am correct.
My conjecture is that all the perfect numbers are constructed by the formula given below or at least the reverse is true i. e. if any number which fits this formula has got to be a perfect number.
2-n x (2-(n+1)-1)
Provided that the number inside the outer set of bracket is a prime number.
I will explain it by example.
If you put 1, 2, and 4 in place of n, you get the first three perfect numbers i. e. 6, 28, and 496.
If the n is 3 the number inside the bracket becomes 15 which is not a prime number so 120 is not a perfect number though incidentally ) it is the first tri perfect number.
Having said that I 'intuitively' know that second part of my conjecture is correct. Or maybe it is not just intuition but something to do with the fact that the factors of the 2-n are 2-(0-n) and nothing else. I would very much welcome if it is contrary to that.
Now if the first part of my conjecture is also correct then all the perfect numbers have to be even. I would very much appreciate the input on this.
reply
Catman
I wonder how we know a potential odd perfect number times 2 would lead to another triperfect
Also something interesting I realized is that if an odd perfect exists, it would have to be a power of a prime, because fhat is one less digit to add to make the sum even again. I think so because all the factors of a nonperfect power of a prime would have an even number of factors, and an even number of odd numbers added together is even, so it wouldn't work (this is the case with odd numbers because all of its factors have to be odd as well. This concludes that the number has to be a power of a prime number, which has an odd number of digits
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I wonder how we know a potential odd perfect number times 2 would lead to another triperfect
Also something interesting I realized is that if an odd perfect exists, it would have to be a power of a prime, because fhat is one less digit to add to make the sum even again. I think so because all the factors of a nonperfect power of a prime would have an even number of factors, and an even number of odd numbers added together is even, so it wouldn't work (this is the case with odd numbers because all of its factors have to be odd as well. This concludes that the number has to be a power of a prime number, which has an odd number of digits
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TriKool
There's something I don't understand. James used the argument that because 2N is triperfect for any odd perfect N, and there probably aren't any odd perfect N, that means there probably aren't any large triperfect numbers. But that's invalid. Not only is it the -converse- inverse of modus ponens (false, it could also be used to show that there likely aren't any triperfect numbers at all, which clearly isn't the case. At 5: 08 he basically said that there is either an odd perfect number -OR- there is no other triperfect number, which doesn't logically follow from the theorem.
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There's something I don't understand. James used the argument that because 2N is triperfect for any odd perfect N, and there probably aren't any odd perfect N, that means there probably aren't any large triperfect numbers. But that's invalid. Not only is it the -converse- inverse of modus ponens (false, it could also be used to show that there likely aren't any triperfect numbers at all, which clearly isn't the case. At 5: 08 he basically said that there is either an odd perfect number -OR- there is no other triperfect number, which doesn't logically follow from the theorem.
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Shaun
Might be related to tangent line angles moving along rough diameter of a circle.
Eg. If there's finite lines and angles and assuming equal (not odd) top number of tangent lines, the angle between each tangent line reaches a limit where cannot find any more fitting numbers inside larger number. Might be a set angle where if go over obtuse? limit between tangent line corners/joins, you cannot find more.
My maths rhetoric is terrible but hopefully it's understood.
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Might be related to tangent line angles moving along rough diameter of a circle.
Eg. If there's finite lines and angles and assuming equal (not odd) top number of tangent lines, the angle between each tangent line reaches a limit where cannot find any more fitting numbers inside larger number. Might be a set angle where if go over obtuse? limit between tangent line corners/joins, you cannot find more.
My maths rhetoric is terrible but hopefully it's understood.
reply
Joshua
Isn't there a logical flaw in this video? It was stated that if N is odd and perfect, then 2N is triperfect, and further that either there is an odd perfect number, or our list of triperfect numbers is complete. But isn't it possible that there are more triperfect numbers, and they just happen not to be twice an odd perfect number?
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Isn't there a logical flaw in this video? It was stated that if N is odd and perfect, then 2N is triperfect, and further that either there is an odd perfect number, or our list of triperfect numbers is complete. But isn't it possible that there are more triperfect numbers, and they just happen not to be twice an odd perfect number?
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Roger
One of the few things that still deeply fascinate and excite me is the fact that, despite the combined brain power of many very clever people over centuries, many very simple maths questions (such as: is there an odd perfect number) cannot be definitively answered. Amazing.
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One of the few things that still deeply fascinate and excite me is the fact that, despite the combined brain power of many very clever people over centuries, many very simple maths questions (such as: is there an odd perfect number) cannot be definitively answered. Amazing.
reply
God
Aside from the Standard Perfect Numbers, the bigger the 'x'perfect number is, the bigger it's total found examples are and I guess the higher it's first example would start. I wonder if there's a chance that a super massive x'perfect could break that chance?
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Aside from the Standard Perfect Numbers, the bigger the 'x'perfect number is, the bigger it's total found examples are and I guess the higher it's first example would start. I wonder if there's a chance that a super massive x'perfect could break that chance?
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-lvaro
do you ever run out of n's? can you find n-perfect numbers for every n? Or it's a finite list as well? If it is, the sum of all n-perfect numbers is finite (or so it's believed, but if it's not, it's infinite. Either way, feels weird somehow.
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do you ever run out of n's? can you find n-perfect numbers for every n? Or it's a finite list as well? If it is, the sum of all n-perfect numbers is finite (or so it's believed, but if it's not, it's infinite. Either way, feels weird somehow.
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turtlellamacow
so what I'm getting from this video is that if -k is N-perfect- means -the sum of the factors of k, including k itself, is Nk-, then perfect numbers should really be called biperfect numbers (and the only perfect number would be one)
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so what I'm getting from this video is that if -k is N-perfect- means -the sum of the factors of k, including k itself, is Nk-, then perfect numbers should really be called biperfect numbers (and the only perfect number would be one)
reply
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