## 21.12.16

### The circle of i

Every day many people around the world look for the next largest Mersenne prime. A Mersenne prime is a Mersenne number that's also prime.

A Mersenne number is an integer of the form 2^i - 1 where i must also be prime but this condition is not strongly enforced in all texts and certainly not on this blog.

I prefer the loose definition where i is not necessarily a prime. With this loose definition in mind, the first few Mersenne numbers are:

0, 1, 3, 7, 15, 31, 63, 127, 255, 511, 1023, 2047, 4095, ...

The first few Mersenne prime numbers are:

3, 7, 31, 127, 8191, 131071, 524287, 2147483647, ...

In a previous post I proved that the order of 2 mod n is equal to i where n is a Mersenne number of the form 2^i - 1

For example, the order of 2 mod 15 is 4 and 15 = 2^4 - 1.

A Mersenne number can also be a product of 2 or more distinct primes.

The first few products of 2 distinct primes that create a Mersenne number are:

5*3 = 15, 7*73 = 511, 23*89 = 2047, 47*178481 = 8388607

At the other end of the spectrum of Mersenne numbers are integers of the form 2^i + 1.

They don't have a fancy name but some of them do show up in fancy parts of Number Theory. I'll call them MersenneSofit numbers.

The first few such numbers are:

2, 3, 5, 9, 17, 33, 65, 129, 257, 513, 1025, 2049, 4097, ...

The only prime MersenneSofit numbers that I was able to find are also Fermat primes even though the set of MersenneSofit numbers is not equivalent to the set of Fermat numbers. I find this quite fascinating although it is possible I missed a prime in my savage late night calculations.

Here are the only MersenneSofit primes I could find so far:

3, 5, 17, 257, 65537

As I already said, these are also the only Fermat primes currently known.

Regardless of whether prime or composite, I claim that the order of 2 mod n for all n of the form 2^i + 1 is 2*i.

A Mersenne Sofit number can also be a product of distinct primes. The first few products of 2 distinct primes are:

5*13 = 65
3*43 = 129
17*241 = 4097
3*2731 = 8193
3*43691 = 131073
3*174763 = 524289
17*61681 = 1048577
3*2796203 = 8388607
17*15790321 = 268435457

Note: although there are more Mersenne primes than MersenneSofit primes, there seem to be less products of 2 distinct primes that form Mersenne numbers than there are products of 2 distinct primes that form MersenneSofit numbers.