## 11.2.17

### A Shrewd Sequence

In previous entries I categorized integers n between 2i and 2i+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 2i - 1), then the order of 2 mod n is equal to i.

2. Then I conjectured that if n is of the form 2+ 1 then the order of 2 mod n is equal to 2*i.

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

2i + (2(i-1) - 3) if i is even
or 2i + (2i-1 + 3) if i is odd

or in other words of the form (here's the proof that these two formulas are equal)

3*(2i-1 - 1) if i is even
3*(2i-1 + 1) if i is odd

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.

Conjecture: For each i > 3 there exists at least one integer n such that 2< n < 2i+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 2i and 2i+1 for each i > 3

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: