The Perfect Number Movie

[Heli Whetzel]

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.

Though the Pythagoreans were interested in the occult properties of Perfect Numbers, they did little of mathematical significance with them. It was around 300 BC when Euclid wrote his Elements that the first real result was made. Although Euclid concentrated on Geometry, many number theory results can be found in his text (Burton, 1980).

If as many numbers as we please beginning from a unit [1] be set out continuously in double proportion, until the sum of all becomes a prime, and if the sum multiplied into the last make some number, the product will be perfect.

This is my code to try and find all the perfect numbers that originate between 1 and 50000000. It works fine for the first 3 numbers, they are between 1 and 10000. But as it progresses it becomes extremely slow. Like maybe going through 1000 numbers every 10 seconds. Then eventually going through 10 numbers every 5 seconds.

2) There are no odd perfect numbers. This is actually a conjecture (not proven yet) but they tried everything below 10^300 and didn't find any. So there are definately exactly none < 50000000. That means you can skip half the terms.

As has already been mentioned, no odd perfect numbers have been found. And according to the Wikipedia article on perfect numbers, if any odd perfect numbers exist they must be greater than 101500. Clearly, looking for odd perfect numbers takes sophisticated methods &/or a lot of time. :)

I'm a grad student in mathematics and I've been working with a very gifted high school student (likely the smartest high school student I've ever met) on problems he's brought up and some competition math problems. This student has developed an interest in perfect numbers and the question regarding existence of odd perfect numbers. He has come up with a conjecture about odd perfect numbers, but I have not studied number theory and hence am not necessarily aware of well-known results of the field. So, here we are.

Perfect numbers are also intimately connected with a class of numbers known as Mersenne primes, which are prime numbers of the form . This can be demonstrated by considering a perfect number of the form where is prime. By definition of a perfect number ,

such that (Eaton 1995, 1996). In addition, all even perfect numbers are hexagonal numbers, so it follows that even perfect numbers are always the sum of consecutive positive integers starting at 1, for example,

In number theory, a perfect number is a positive integer that is equal to the sum of its positive factors, excluding the number itself. The most popular and the smallest perfect number is 6, which is equal to the sum of 1, 2, and 3. Other examples of perfect numbers are 28, 496, and 8128.

A perfect number is a positive integer that is equal to the sum of its factors except for the number itself. In other words, perfect numbers are the positive integers that are the sum of its divisors. The smallest perfect number is 6, which is the sum of its factors: 1, 2, and 3. It is to be noted that this sum does not include the number itself which is also a factor of itself.

Do you know that when the sum of all the divisors of a number is equal to twice the number, the number has a separate name? Such numbers are called complete numbers. In fact, all the perfect numbers are also complete numbers.

The initial study of perfect numbers may go back to the Egyptians who might have come across such numbers naturally. There is not much information regarding the discovery of perfect numbers. It is said that perhaps the Egyptians may have discovered them. Despite knowing the existence of perfect numbers, it was only the Greeks who were eager to study more about these numbers. Perfect numbers were studied by Pythagoras and his followers for its mystical properties. The smallest perfect number found was 6. This number 6 gathered much attention in the beginning by the Pythagoreans, more for its mystical and numerological properties than for any mathematical significance. It is to be noted that 6 is the smallest perfect number, the next being 28.

In order to find a perfect number, we can use the technique told by Euclid. According to Euclid, there is an expression that can be a perfect number subject to a specific condition. According to his proposition, if 2n -1 is a prime number, then 2n-1(2n-1) is a perfect number. This condition can be understood using the following table. Euclid said that (2n - 1) multiplied by 2n - 1, can be a perfect number if the term in the bracket, that is, (2n - 1) is a prime number. In other words, [2n - 1 (2n - 1) = perfect number], if (2n - 1) is a prime number.

Therefore, we need to find a value of 'n' for which (2n - 1) is prime. So, the following table will help us understand this better. Let us follow the steps given below so that we can relate to the table and understand the process.

A few perfect numbers 6, 28, 496 and 8128 are known to us since ancient times. Let us see their divisors and their sum through the table given below. Their sum results in the number itself. Therefore, these are known as perfect numbers.

A perfect number is a positive integer that is equal to the sum of its factors excluding the number itself. For example, 6 is a perfect number because when we add all its factors except 6, we get, 1 + 2 + 3 = 6. We get the sum as the number itself. Therefore, 6 is a perfect number.

A perfect square is a number, which can be expressed as the square of a number from the same number system whereas a perfect number is a number, which can be expressed as the sum of its factors except itself.

If you can factor a number into all of its divisors, you can immediately add them all up and discover, for yourself, whether your number is perfect or not. For the first few numbers, this is a straightforward task, and you can see that most numbers aren't perfect at all: they're either abundant or deficient.

You might look at these numbers, the ones that happen to be perfect, and start to notice a pattern here as to how these numbers can be broken down. They're all the result of multiplying 2 to some power, let's call it X, by a prime number. And interestingly, the prime number you're multiplying it by is always equal to one less than double what 2X is.

If you do a search for perfect numbers up to 10,000 you will find only the following perfect numbers: \[\beginaligned 6 &=2\cdot 3, \\ 28 &=2^2\cdot 7, \\ 496 &=2^4\cdot 31, \\ 8128 &=2^6\cdot 127.\endaligned\] Note that \(2^2=4\), \(2^3=8\), \(2^5=32\), \(2^7=128\) so we have: \[\beginaligned 6 &=2 \cdot (2^2-1), \\ 28 &=2^2\cdot(2^3-1), \\ 496 &=2^4\cdot(2^5-1), \\ 8128 &=2^6\cdot(2^7-1).\endaligned\] Note also that \(2^2-1\), \(2^3-1\), \(2^5-1\), \(2^7-1\) are Mersenne primes. One might conjecture that all perfect numbers follow this pattern. We discuss to what extent this is known to be true. We start with the following result.

Write \(q=2^p-1\) and let \(n=2^p-1q\). If \(q\) is a Mersenne prime, then since it is odd and prime, by Theorem 1.15.1(2) we have \(\sigma(n)=\sigma\left(2^p-1q\right)=\left(\frac2^p-12-1\right) \left(\fracq^2-1q-1\right)=(2^p-1)(q+1) = (2^p-1)2^p = 2n\). That is, \(\sigma(n) = 2n\) and \(n\) is perfect.

Let \(n\) be even and perfect. Since \(n\) is even, \(n=2m\) for some \(m\). We take out as many powers of \(2\) as possible, obtaining \[n=2^k\cdot q,\quad k\ge 1\text, $q$ odd.\nonumber \] Since \(n\) is perfect, \(\sigma^\ast(n)=n\), that is, \(\sigma(n)=2n\). Since \(q\) is odd, \(\gcd(2^k,q)=1\), so by Lemmas 1.15.1 and 1.15.2, \[\sigma(n)=\sigma(2^k)\sigma(q)=(2^k+1-1)\sigma(q).\nonumber \] So we have \[2^k+1q=2n=\sigma(n)=(2^k+1-1)\sigma(q),\nonumber \] hence \[\labeleq: thm 16-2 first 2^k+1q=(2^k+1-1)\sigma(q).\] Now \(\sigma^\ast(q)=\sigma(q)-q\), so \[\sigma(q)=\sigma^\ast(q)+q.\nonumber \] Putting this in equation \(\eqrefeq: thm 16-2 first\) we get \[2^k+1q=(2^k+1-1)(\sigma^\ast(q)+q)\nonumber \] or \[2^k+1q=(2^k+1-1)\sigma^\ast(q)+2^k+1q-q,\nonumber \] which implies \[\labeleq: thm 16-2 second \sigma^\ast(q)(2^k+1-1)=q.\] In other words, \(\sigma^\ast(q)\) is a divisor of \(q\). Since \(k\ge 1\) we have \(2^k+1-1\ge 4-1=3\). So \(\sigma^\ast(q)\) is a proper divisor of \(q\). But \(\sigma^\ast(q)\) is the sum of all proper divisors of \(q\). This can only happen if \(q\) has only one proper divisor. This means that \(q\) must be prime and \(\sigma^\ast(q)=1\). Then equation \(\eqrefeq: thm 16-2 second\) shows that \(q=2^k+1-1\). So \(q\) must be a Mersenne prime and \(k+1=p\) is prime. So \(n=2^p-1(2^p-1)\), as desired.

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