^{i}and 2

^{i+1}where the order of 2 mod n has a common formula for each i.

For example:

1. I proved that if n is a Mersenne number (an integer of the form 2

^{i}- 1), then the order of 2 mod n is equal to i.

2. Then I conjectured that if n is of the form 2

^{i }+ 1 then the order of 2 mod n is equal to 2*i.

3. I also conjectured that if n is of the form:

2

^{i}+ (2

^{(i-1)}- 3) if i is even

or 2

^{i}+ (2^{i-1}+ 3) if i is odd
or in other words of the form (here's the proof that these two formulas are equal)

then the order of 2 mod n is i + (i - 2)

Another example that I have not previously published on this blog is also based on just a conjecture that may turn out to be false.

Below is a sequence generated using the first occurrence of such integers between 2

21, 35, 75, 231, 301, 731, 1241, 2079, 7513, 8337, 16485, 39173, 66591, 131241, 371365, 539973, 1125441, 2153525,...

Below is a slow calculator for finding the order of 2 mod n for any odd integer n:

3*(2

3*(2

^{i-1}- 1) if i is even3*(2

^{i-1 }+ 1) if i is oddthen the order of 2 mod n is i + (i - 2)

Another example that I have not previously published on this blog is also based on just a conjecture that may turn out to be false.

**Conjecture:**For each i > 3 there exists at least one integer n such that 2^{i }< n < 2^{i+1}and the order of 2 mod n is equal to (i-1)*(i-2)Below is a sequence generated using the first occurrence of such integers between 2

^{i}and 2^{i+1}for each i > 321, 35, 75, 231, 301, 731, 1241, 2079, 7513, 8337, 16485, 39173, 66591, 131241, 371365, 539973, 1125441, 2153525,...

Below is a slow calculator for finding the order of 2 mod n for any odd integer n: