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2015-09-09, 13:23   #22
T.Rex

Feb 2004
France

16328 Posts

Quote:
 Originally Posted by Batalov If you eliminate more and more, finally you simply get yourself in an illusory environment when you can no longer find counterexamples, -- and so what? It simply means that no one is willing to find them, but they are still there, only larger. .... After being presented with acorns, he further "improves" his theory to 'Tiggers eat evertyhing except honey and acorns'. Can you blame anyone for not taking him all too seriously after that?
However, don't you think that there are other ways for finding new stuff than using pure Math proof ? Is there a place for experimenting ? Years ago, before computers were there, many mathematicians used pen, paper, and figures, for exploring something that looked weird for them at first, before they search for a math proof. Now, we have computers. Remember S. Ramanujan: he never used more than a slate since he was unable to pay for note-books and paper at first and because he had a wrong idea about what "good" maths are: find a property and PROVE it ; however his findings were remarkable and mathematicians are still studying his findings. However, there was only 1 Srinivasa...

Now, back to what I did:
- primus published 4 conjectures about a property of primes
- I generalized the conjectures into one
- someone proved the conjecture and, as you explained, it was not so difficult to prove and it is simply another way to present something already known
- however, what about pseudoprimes ?
- I found a limited number of pseudoprimes
- I used the idea: not only S(n-1) must have a special value Sn, but S(i) must not be equal to S(n) when i<n-1, because some Sn values are a dead-end, like 2 : 2^2-2=2 , and because we must not enter a cycle with length smaller than n-1, with the idea that a minimum of n-1 different steps are required for showing a prime, as for Mersenne/Fermat LLT. The other idea is that, for very simple numbers like Mersenne and Fermat, so close to 2^n, about n iterations of X^2-2 mod N is enough, though for numbers moving away from pure 2^n more steps should be required, which are the P_k and P(c) functions that add more steps before and after the (s^2-2 mod N) part. The idea is that there must be a minimum of steps to run in order to have something that can be a prime test. A number N being a prime creates a symmetric structure ; running a minimum number of steps in the structure of a number is required in order to discover if the symmetry is broken due to a factor of N. What is magic with LLT for Mersenne/Fermat is this: you run n-1 steps in the digraph, from a starting point to a ending point, and bingo! it is a prime. This should work, under more complex conditions, for other numbers, where starting and ending points are more difficult to define since they depend on other characteristics of the number than the power of n : k and c in addition, like primus did.
- then, I found that ALL the remaining pseudoprimes I can find (c>0) share the same characteristics: c= 1 mod 8 , and N=0 mod (2*c-1) .
- and then I shown that ALL these remaining pseudoprimes look strange: only 1 small factor (2c-1) and a big prime. They do NOT look like random factors as expected for random exceptions in the theory. They share a property, probably showing something deep and interesting to look at.
- and, at the end, filtering this unique class of pseudoprimes, I cannot find more pseudoprimes (up to the limits of my computer).

For sure, that does NOT provides a proof.
That is only showing "strange" properties that seem to exist with very big numbers and with a very big number of numbers tried.
And, as you say, after honey and acorns, there could be ants and mushrooms. Or not.

I remember an example from a book of JP Delahaye, where 2 sequences A(n) and B(n) generate coprime numbers ... up to n being very very big where an exception occured.

But, in our case, we are using methods that are well known and have already provided good results, like LLT for Mersenne numbers and LLT for Fermat numbers, which are true prime tests.
So, we have the expectation to find something that works, under strict conditions.

So, I have only used 2 filters.

Remind also that, for c<0, I found that all pseudoprimes I've found have the same structure.

For sure, there could be another kind of pseudoprimes with much higher values for: k, n, or c . And, in that case, the conjecture will be the Tiger you are talking about.

Last fiddled with by T.Rex on 2015-09-09 at 14:02 Reason: english