Perfect Number (2012) Full Movie Download
Gabriel Litke
In number theory, a perfect number is a positive integer that is equal to the sum of its proper positive divisors, that is, the sum of its positive divisors excluding the number itself (also known as its aliquot sum). Equivalently, a perfect number is a number that is half the sum of all of its positive divisors (including itself) i.e. σ1(n) = 2n.
In Book IX of The Elements, Euclid gave a method for constructing perfect numbers (Dickson 2005, p. 3), although this method applies only to even perfect numbers. In a 1638 letter to Mersenne, Descartes proposed that every even perfect number is of Euclid's form, and stated that he saw no reason why an odd perfect number could not exist (Dickson 2005, p. 12). Descartes was therefore among the first to consider the existence of odd perfect numbers; prior to Descartes, many authors had implicitly assumed (without proof) that the perfect numbers generated by Euclid's construction comprised all possible perfect numbers (Dickson 2005, pp. 6-12). In 1657, Frenicle repeated Descartes' belief that every even perfect number is of Euclid's form and that there was no reason odd perfect number could not exist. Like Frenicle, Euler also considered odd perfect numbers.
To this day, it is not known if any odd perfect numbers exist, although numbers up to have been checked without success, making the existence of odd perfect numbers appear unlikely (Ochem and Rao 2012). The following table summarizes the development of ever-higher bounds for the smallest possible odd perfect number.
More recently, Hare (2005) has shown that any odd perfect number must have 75 or more prime factors. Improving this bound requires the factorization of several large numbers (Hare), and attempts are currently underway to perform these factorizations using the elliptic curve factorization method at mersenneforum.org and OddPerfect.org. Ochem and Rao (2012) subsequently showed that any odd perfect number has at least 101 not necessarily distinct prime factors.
6 10 = 2 2 + 2 1 = 110 2 28 10 = 2 4 + 2 3 + 2 2 = 11100 2 496 10 = 2 8 + 2 7 + 2 6 + 2 5 + 2 4 = 111110000 2 8128 10 = 2 12 + 2 11 + 2 10 + 2 9 + 2 8 + 2 7 + 2 6 = 1111111000000 2 \displaystyle \beginarrayrcl6_10=&2^2+2^1&=110_2\\28_10=&2^4+2^3+2^2&=11100_2\\496_10=&2^8+2^7+2^6+2^5+2^4&=111110000_2\\8128_10=&\!\!2^12+2^11+2^10+2^9+2^8+2^7+2^6\!\!&=1111111000000_2\endarray
It is unknown whether any odd perfect numbers exist, though various results have been obtained. In 1496, Jacques Lefvre stated that Euclid's rule gives all perfect numbers,[17] thus implying that no odd perfect number exists. Euler stated: "Whether ... there are any odd perfect numbers is a most difficult question".[18] More recently, Carl Pomerance has presented a heuristic argument suggesting that indeed no odd perfect number should exist.[19] All perfect numbers are also harmonic divisor numbers, and it has been conjectured as well that there are no odd harmonic divisor numbers other than 1. Many of the properties proved about odd perfect numbers also apply to Descartes numbers, and Pace Nielsen has suggested that sufficient study of those numbers may lead to a proof that no odd perfect numbers exist.[20]
All even perfect numbers have a very precise form; odd perfect numbers either do not exist or are rare. There are a number of results on perfect numbers that are actually quite easy to prove but nevertheless superficially impressive; some of them also come under Richard Guy's strong law of small numbers:
The sum of proper divisors gives various other kinds of numbers. Numbers where the sum is less than the number itself are called deficient, and where it is greater than the number, abundant. These terms, together with perfect itself, come from Greek numerology. A pair of numbers which are the sum of each other's proper divisors are called amicable, and larger cycles of numbers are called sociable. A positive integer such that every smaller positive integer is a sum of distinct divisors of it is a practical number.
A semiperfect number is a natural number that is equal to the sum of all or some of its proper divisors. A semiperfect number that is equal to the sum of all its proper divisors is a perfect number. Most abundant numbers are also semiperfect; abundant numbers which are not semiperfect are called weird numbers.
The odd perfect number problem likely needs no introduction. Recent progress (where by recent I mean roughly the last two centuries) seems to have focused on providing restrictions on an odd perfect number which are increasingly difficult for it to satisfy (for example, congruence conditions, or bounding by below the number of distinct prime divisors it must have). By reducing the search space in this manner, and probably due to other algorithmic improvements (factoring, parallelizing, etc.), there has also been significant process improving lower bounds for the size of such a number. A link off of oddperfect.org claims to have completed the search up to $10^1250$.
But, assuming my admittedly cursory reading of the landscape is correct, none of the current research seems particularly equipped to prove non-existence. The only compelling argument I've seen on this front is "Pomerance's heuristic" (also described on oddperfect.org). Worse, and maybe this is really the point of this question, it would be a little disappointing if the non-existence proof was an upper bound of $10^1250$ (depending on the techniques used to get the bound) combined with the above brute force search.
Aside: I hope this does not come off as dismissive of "elementary" techniques, or of the algorithmic ones mentioned in the first paragraph. Indeed, they have, to my knowledge, been the only source of progress on this problem, and certainly contain interesting mathematics. Rather, this phrasing stems from my desire to find anything in the intersection of "odd perfect number theory" and "things I know anything about," and perhaps a desire to see the odd perfect number problem settled without the use of a beyond-gigantic brute force search.
This is a problem I have thought alot about. I have not seen any of the modern techniques in your list applied to the problem. Part of the issue is that if you represent $\sigma(n)=2n$ as a Diophantine equations in $k$ variables (corresponding to the prime factors--but allowing the powers to vary) then there are lots of solutions (just not where all the variables are simultaneously prime). So the usual methods of trying to show non-existence of solutions just don't cut it. Historically, this multiplicative approach is the one many people have taken, because at least some progress can be made on the problem. My personal feeling is that maybe someday these bounding computations will be tweaked to the point that they lead to the discovery of some principle that will solve the problem. For example, in one of my recent papers, I was led to consider the gcd of $a^m-1$ and $b^n-1$ (where $a$ and $b$ are distinct primes). I would conjecture that this gcd has small prime factors unless $m$ or $n$ is huge. If that happens, many of the computations related to bounding OPNs become much easier.
I agree with Pace - the correct function to consider would be the abundancy index instead of the sigma function itself. In a certain sense, the abundancy index value of 2 for perfect numbers (odd or even) has served as a baseline on which various other properties and concepts related to number perfection were developed.
Though I have seen modular forms before (having done a [modest] exposition of elliptic curve theory and related topics) for my undergraduate thesis, I have likewise not seen any 'objects' used in Wiles' FLT proof applied directly to the OPN conjecture -- at least, not 'directly' in the literal sense of the word.
I had an opportunity to work with Dr. Beauregard Stubblefield 35 years ago (yep, 62 and proud of it) and he generously gave the credit for the group's result to Dr. Mary Buxton and me. Dr. Stubblefield of course did most of the work, and he and I discussed many things. It seems that Leopold Kronecker, the great German algebraist 1823-1891, had it right the whole time. He proved that X = p^e + p^(e+1) + ... + p^2 + p + 1, where e is one less than a prime and p is a prime, cannot be algebraically reduced. But we knew that. The big deal is that the numeric factors of the expression are either (e + 1) X or (k(e + 1) + 1) X. Dr. Stubblefield found the pattern and I found that Kronecker had proved it. Stubblefield used the result in his Proposition 11, which he proved. Thirty-five years ago we used the result to factor many sigma(p^e). We talked about extending the result back then so we both had deep input into the new theory. Recently, after an engineering career in the auto industry in Detroit, I extended Proposition 11 to apply to many more cases in several different ways. I think, Cam, that this is the road you are suggesting. Steve Elmore
The odd perfect number problem does have a connection to modular forms. the divisor funct can be written as a function of the tau function and sigma_k(n) = sum_n d^k. The earlier example is the van der Pol identity. This was used by Touchard to conclude that n = 36a + 9 or 12b + 1.
But four, I thought, four was the perfect number. The older ones can pair off or the younger ones can or they can all be friends. The likelihood that you end up with a solo act in there was certainly lessened.
Now I look at my kids and my family. There are four of them. We were all boys in my family and now I have the one girl mixed in. Also, I never considered the idea of multiples when I was building my theoretically perfect family of four kids.
Clearly the twins, while not at all identical, will have a bond beyond that of normal, separate-birth siblings. This for no other reason then they will always be in the same grade and will, in all likelihood, spend the next sixteen years or so going to the same school together every day.
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