Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Mathematics of radio engineering wikipedia , lookup
Infinitesimal wikipedia , lookup
Georg Cantor's first set theory article wikipedia , lookup
List of prime numbers wikipedia , lookup
Large numbers wikipedia , lookup
Real number wikipedia , lookup
Proofs of Fermat's little theorem wikipedia , lookup
Determining if any Odd Perfect Numbers Exist. CORYDEN ENOCHS βThe world was created in 6 days and the moon orbits in 28 days because 6 and 28 are perfect.β βPhilo of Alexandria What is a perfect number? A number N is perfect if it is equal to twice the sum of its divisors: π π = 2π. For example: 6 is the first perfect number. Its positive divisors are 1, 2, 3, and 6. So, π π = 12 and 2π = 12. The second perfect number is 28. π π = 1 + 2 + 4 + 7 + 14 + 28 = 56 and 2π = 56. The list of perfect numbers continues with 496,8128,33550336,8589869056,... Another view of perfect numbers is to recognize that the sum of the positive proper divisors is the original number. Some History The definition of perfect numbers was first found in Euclidβs Elements. The first four perfect numbers are considered to have been discovered by Euclid. The fifth was seen referenced in a manuscript written by an unknown mathematician between 1456-1461. Pietro Cataldi showed the sixth and seventh perfect numbers in 1588. Even Perfect Numbers As we have seen, the list of perfect numbers contains only even numbers so far. It was proved by Euclid that 2πβ1 2π β 1 is an even perfect number whenever (2π β 1) is prime. We can see the first four generated in this form with p=2,3,5,7 p=2: 21 22 β 1 = 2 4 β 1 = 2 β 3 = 6 p=3: 22 23 β 1 = 4 8 β 1 = 4 β 7 = 28 p=5: 24 25 β 1 = 16 32 β 1 = 16 β 31 = 496 p=7: 26 27 β 1 = 64 128 β 1 = 64 β 127 = 8128 Prime Numbers of the form 2π β 1 are known as Mersenne Primes. In the 18th century, Euler proved that this formula will yield ALL even perfect numbers. This is known as the Euclid-Euler Theorem. How Many are There? There is a one-to-one relationship between Mersenne Primes and perfect numbers, so each Mersenne Prime generates one perfect number. As of April 2015, 48 Mersenne Primes have been discovered, so there are currently 48 known perfect numbers. It is not yet known if there are infinitely many perfect numbers, or if there are infinitely many Mersenne Primes. In other words, there could potentially be a ceiling, after which no other perfect numbers exist. But I will leave this to another discussion. Odd Perfect Numbers While there are no known odd perfect numbers, and it has been conjectured that none even exist, there have been a variety of results regarding necessary conditions that must hold true of a perfect odd number. One such result is a lower bound for possible solutions. This is currently 101500 . This is simply an immensely large number as a lower bound for any possible odd perfect numbers. Another result concerns the number of distinct prime factors. This has currently been proven to require at least 10 distinct prime factors. If not considering distinctness, the number of prime factors has gone up as high as 101 prime factors. It has also been shown that the greatest prime factor must be greater than 108 , the second largest greater than 104 , and the third largest greater than 100. 1500 10 100000000000000000000000000000000000000000000000000000000000000000000000000000000000000000 000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000 000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000 000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000 000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000 000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000 000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000 000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000 000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000 000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000 000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000 000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000 000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000 000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000 000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000 000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000