## Numbers, Mersenne Primes and Fermat Primes

### Mersenne Primes

A Mersenne prime is a prime that is one less than a power of 2.
Examples include 3, 7, and 31.
The exponent on a Mersenne prime must also be prime.
To illustrate, consider 215-1.
Now 15 is not prime, infact it is 3×5,
so replace 23 with 8, and write 85-1.
This is divisible by 8-1,
just as xn-1 is divisible by x-1.

If p is a Mersenne prime, say 2k-1,
then consider n = p×2k-1.
Let's compute σ(n).
There is one instance of p, so the first component of σ is 1+p.
We know this better as 2k.
The second factor of n is 2, possibly raised to a high power.
Hence the second component of σ becomes 1+2+4+8+16+…+2k-1.
This is equal to 2k-1.
We know this better as p.
Therefore σ(n) = 2k×p, which is twice n, hence n is a perfect number.
Perfect numbers include 6, 28, and 496, corresponding to the first three Mersenne primes.

conversely, let n be a perfect even number.
Let n contain k-1 powers of 2.
Thus σ(n) includes 2k-1, which we will call j.
If σ(n) = 2n, then j, an odd number, also divides n.
If j is prime, it contributes 1+j, or 2k, to σ(n).
This makes j a Mersenne prime,
and n is a perfect even number as described above.
If j is anything else, a power of a prime,
or a product of multiple primes, σ(j) exceeds j+1.
With σ(j) > 2k, n becomes abundant.
Any additional factors, besides 2 and j, will also make n abundant.
Therefore all perfect even numbers have been characterized.
They correspond one to one with the Mersenne primes.

No perfect odd numbers have been found.

A Fermat prime is one greater than a power of 2.
Examples include 3, 5, 17, 257, and 65537.
Consider the number 2(2h×n)+1,
where n is odd, and n > 1.
If x is 2(2h),
then our fermat prime can be written as xn+1,
which is divisible by x+1.
To be prime,
n must equal 1,
whence the exponent is a power of 2.
Review the 5 examples above.
Each is 2 to a power of 2, plus 1.
In fact, these are the only known Fermat primes,
and we believe there are no others.