The values 1 and 8, suggest that the formula can be something like this:

2^n * x, where x is = 1 OR 2^n + x, where x is = 0

Value 1 is in the position 0, value 8 is in the position 3.

Position 0 has the value 1, which is 2^0*1 OR 2^0+0

Position 3 has the value 8, which is 2^3*1 OR 2^3+0

It means that the calculation of x for the positions 0 and 3 result in 0 or 1.

Lets see if that's right...

Position 1 has the value of

**3**, which is 2

^{1}*1 =

**2** OR 2

^{1}+0 =

**2**Hmm. That didn't work.

Position 2 has the value of

**7**, which is 2

^{2}*1 =

**4** OR 2

^{2}+0 =

**4**Hmm. That didn't work either.

Position 4 has the value of

**21**, which is 2

^{4}*1 =

**16** OR 2

^{4}+0 =

**16**Hmm. Still no good.

Position 5 has the value of

**49**, which is 2

^{5}*1 =

**32** OR 2

^{5}+0 =

**32**This isn't looking like it's going to work.

Lets think about this logically.

If (as many of us have presumed) the first key (position 0) is completely random in the range between 2

^{0} and 2

^{1}-1 then all the possible values are 1 and, well, um, I guess that's it. Can't be anything else.

If the third key is then completely random in the range between 2

^{3} and 2

^{4}-1, then there are 4 possibilities 5, 6, 7, and 8. So there's a 25% chance of it being 8.

If you just pick any 2 of the lower positions the possible values are so limited that it's pretty easy to imagine that you see patterns in the numbers. The problem is none of the patterns people seem think they see work out as soon as you add another position or two. If you're having to come up with a brand new "formula" to fit the data every time you add a position, then its important to consider the possibility that you are looking at random data and simply calculating a convoluted and useless formula that fits your current limited set and none of the rest of the set.

The source also pointed out this example:

If you calculate the Prime Factor of 21 (Position 4), the result is 3 and 7. (these numbers are in the position 1 and 2).

Prime factors of 76 are 2 and 19 (which don't occur in any positions).

Prime factors of 224 are 2 and 7. Now 7 was already used as a factor for 21, and 2 is a factor of 76 and doesn't occur in the list of positions.

Prime factors of 467 are 467 and that's it. (467 is prime).

If you like I can give you a formula that will work "perfectly" for 39 and 40 (mostly because 2 points define a line and it's pretty easy to create an equation for a line). It won't work for pretty much any others, but then all we have to do is figure out how to "tweak" the formula, right?

If you like, I can give you a formula that will work "perfectly" for 38, 39, and 40 (mostly because 3 points define a parabola, and it's pretty easy to create an equation for a parabola). It won't work for pretty much any others, but then all we have to do is figure out how to "tweak" the formula, right?

Here's a formula that I suspect will work EVERY time:

Given zero based position n

Private key k will always be:

k = 2

^{n} + x

Where x will always be in the range between 0 and 2

^{n}x will be different every time (and almost certainly random), but once you figure out x (through brute force) at any position, you'll have the private key at that position.

Another way to look at it is that k will always be random but limited to a random number in the range:

2

^{n} <= k < 2

^{n+1}