A strange idea......

[epatka]

Well, we could start testing factors with 100 bits and work upward. The factors will be much smaller than the huge mersenne numbers they divide.

I guess you have experience in this. :)

The program calculates 1/F using long division just like it's done by hand, except it uses binary.
For example, F = 97 = 1100001:

........

Now we are counting to millions instead of storing millions of bits.

Well, I would lie if I said I understand all of it, but I think I can see the whole picture.

It doesn't sound bad so far......

Look at my first post on page 2 of this topic.
In the second Code: block, the numbers in the "order" column are the width in bits of the repeating pattern. Notice that when the pattern is an even number of bits wide, you can split it in 2 and the halves will add to 11111... When it's an odd number of bits wide, you can't split it anywhere and have it add to 1111....
I think this is true for your decimal examples as well, and for any base.

I see. I have a feeling, there must be a pattern to those too, but it is probably very hard to find :(

What I've been thinking was that maybe we could do the division until the remainder is 1, and that would mean, the number is almost certainly a prime.
If you took a look at my links, you have probably seen that if the composite numbers have a pattern, in front of the pattern there is one or more digits, and ONLY THEN does the pattern start.

For example 70:

0 142857 142857 142857 142857

But most of these numbers are even numbers anyway.

39 has a pattern, but it adds up to 6, not 9. etc.

But prime numbers don't have anything before the pattern.....
Because of this, we could do the division, and watch for the point when the remainder is 1.

I show you what I think.

Here is No. 79
It is prime, and the residue has a pattern, but it doesn't add up to 9:

0.0126582278481

Here is the good old fashioned way of dividing 1 by 79 :)
Under the division the residues. (I hope it will look OK.)

1/79 = 0.0126582278481
100
_210
__520
___460
____650
_____180
______220
_______620
________670
_________380
__________640
____________80
______________1

I thought we could watch for the remainder of 1. That wouldn't be too bad, because the residue is most of the time smaller than the number itself. It might be twice as much work, but it still could be done. I hope :)

This is exciting, even if it doesn't work out ;)


Eva

[epatka]

Just to add something:

As far as I can see, the sum of the digits of the repeating part of the prime numbers is always divisible by 9, even if the digits don't add up to 9 pairwise. That's why I think we might be able to find a general rule for all the primes.

Other composits don't have this quality. Or their digits adds up pairwise to 6, 8, etc.

There is a definite regularity to the primes.

Eva

[David John Hill Jr]

I like to look at primes in terms of Wilson,s theorem where to
a factorial +1 , a number that is indivisible , multiplies,
as perhaps the place to look for the inverse of a prime, as a specific.

[epatka]

It sounds interesting, but I must admit I don't know anything about this.
Would you mind explaining it in a few words?

[cheesehead]

Originally Posted by epatka

Hmmmmm, how do I get rid of this double post?

Ask a forum moderator (philmoore or ewmayer) or Xyzzy to delete it.