MERSENNE NUMBERS
The key to constructing perfect numbers is a collection of numbers named after Father Marin Mersenne, a French monk who studied at a Jesuit college with René Descartes.
Father Marin Mersenne
Both men were interested in finding perfect numbers. Mersenne numbers are constructed from powers of 2, the doubling numbers 2, 4, 8, 16, 32, 64, 128, 256, . . ., and then subtracting a single 1.
A Mersenne number is a number of the form 2 n − 1. While they are always odd, they are not always prime. But it is those Mersenne numbers that are also prime that can be used to construct perfect numbers. Mersenne knew that if the power was not a prime number, then the Mersenne number could not be a prime number either, accounting for the non-prime powers 4, 6, 8, 9, 10, 12, 14 and 15 in the table.
The Mersenne numbers could only be prime if the power was a prime number, but was that enough ?
For the first few cases, we do get 3, 7, 31 and 127, all of which are prime. So is it generally true that a Mersenne number formed with a prime power should be prime as well ?
Many mathematicians of the ancient world up to about the year 1500 thought this was the case. But primes are not constrained by simplicity, and it was found that for the power 11 (a prime number), 2 11 – 1 = 2047 = 23 × 89 and consequently it is not a prime number. There seems to be no rule.
The Mersenne numbers 2 17 – 1 and 2 19 – 1 are both primes, but 2 23 – 1 is not a prime, because
The key to constructing perfect numbers is a collection of numbers named after Father Marin Mersenne, a French monk who studied at a Jesuit college with René Descartes.
Father Marin Mersenne
Both men were interested in finding perfect numbers. Mersenne numbers are constructed from powers of 2, the doubling numbers 2, 4, 8, 16, 32, 64, 128, 256, . . ., and then subtracting a single 1.
A Mersenne number is a number of the form 2 n − 1. While they are always odd, they are not always prime. But it is those Mersenne numbers that are also prime that can be used to construct perfect numbers. Mersenne knew that if the power was not a prime number, then the Mersenne number could not be a prime number either, accounting for the non-prime powers 4, 6, 8, 9, 10, 12, 14 and 15 in the table.
The Mersenne numbers could only be prime if the power was a prime number, but was that enough ?
For the first few cases, we do get 3, 7, 31 and 127, all of which are prime. So is it generally true that a Mersenne number formed with a prime power should be prime as well ?
Many mathematicians of the ancient world up to about the year 1500 thought this was the case. But primes are not constrained by simplicity, and it was found that for the power 11 (a prime number), 2 11 – 1 = 2047 = 23 × 89 and consequently it is not a prime number. There seems to be no rule.
The Mersenne numbers 2 17 – 1 and 2 19 – 1 are both primes, but 2 23 – 1 is not a prime, because
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