Joao Silva wrote:
> All the "transformed Fermat numbers" are primes!
So you say, but I'm sorry to say that I have limited trust in you by now.
> I have devised a special-purpose factoring algorithm which
> was recently expanded into a general purpose one...
> Unfortunately, I do not plan to reveal it here. Let's just say
> that if anyone is able to find a factor for any of the
> "transformed Fermat numbers" I will gladly leave the group
> and you will never here from me again.
The numbers are prp's (probable primes). They are almost certainly
prime but I don't believe you have proved primality of the largest.
> After all, I vividly
> remember my "naive mistake" with the 60229 decimal
> digit PRP that I submitted some months ago...
Actually, that was the 60239-digit composite repunit (10^60239-1)/9
you submitted to the Prime Pages. It has the small factor 4073791
so Chris didn't have to make a prp test.
If you don't know the difference between a prp and a proven prime
then look e.g. at
http://primes.utm.edu/prove/proving.html
> You are right, rule is not the right word! It is simply an
> interesting mathematical observation....
Glad to see you admit "rule" was bad. We also disagree
on "interesting" here, but that's more subjective.
> F5 = 4294967297 = 641* 6700417 but
> 4294969297 = 4294967297 + 2 *10^3 = 643 * 6679579
> which gives two factors of similar size and same order of
> magnitude.
Now you are changing a digit by more than 1 (and you may
have tried with other numbers than F5). This gives so many
possibilities that it is not very surprising you can find a small
number with a similar factorization.
If you find what you think is a remarkable property then you
should really consider whether it would just be an
unsurprising event for random numbers. It would be more
impressive if you found a similar property for
F7 = 59649589127497217 * 5704689200685129054721
The law of small numbers may be of interest:
http://primes.utm.edu/glossary/page.php?sort=LawOfSmall
> Finally, I did not understand what you meant by (probably
> because everybody here writes in a very abbreviated manner?!):
>
> A better "rule" is:
> There is an n-digit near-repdigit prime for each n>1.
> Heuristic excercises:
> Do you expect exceptions?
> Would you search for them?
This is only vaguely related to your posts and you can ignore it.
A near-repdigit prime is a prime where all digits except one
are the same, e.g. 3333373.
I disliked your use of the word rule. My "rule" is about something
else where it looks completely infeasible to find exceptions.
However, I do expect very rare exceptions. I would even conjecture them.
Andersen's near-repdigit conjectures:
1) There are infinitely many n without an n-digit near-repdigit prime.
2) No such n will ever be found.
And congratulations to Harvey Dubner on retaking the palindromic prime record:
http://primes.utm.edu/top20/page.php?id=53
--
Jens Kruse Andersen