- View SourcePredrag,

Thank you.

(Sorry if this seems off-topic and seems like poetry - it probably is) Carol G. Kirnon is my best female friend in the whole wide world. She was the first girl to steal my heart when we were in high school. Therefore, since math is my love and she is my love, I named the first set of numbers after her. The second set, and really the second set, because I encountered them days after is named for the baby girl that had the greatest inpact on my life so far, Kynea R. Griffith (she is ten years old now). I hope some day when people talk about Carol and Kynea numbers, they will know a little bit about the two. Hopefully soon, I will include a biography of the two on Steven Harvey's site.

-----Cletus Emmanuel

pminovic <pminovic@...> wrote:

Cletus,

Sorry, I was not aware of your GC, but I found it now at the bottom

of Steven Harvey's CK page. By the way, the comment "Generalized

Carol" was attached to Ballinger's base 6 GC when I submitted my

prime. At the time it was already confirmed to be prime. I guess the

editor of the site removed the comment manually since he doesn't

approve such "unofficial" comments (?).

Bt the way, who are Carol and Kynea and why are these forms named

after them?

And congrats on your large PRP, (2^148330+1)^4-2, the largest

reported so far.

Regards,

Predrag

--- In primenumbers@yahoogroups.com, Cletus Emmanuel <cemmanu@y...>

wrote:> Congrats to the two of you on your findings. I went to the Top-

5000 and did not see the comments next to either number....

> There is Generalized Carol Number in the form ---> GC = (273*2^k -

1)^2 - 2. This form is more dense than the regular Carol Number. GC

is prime for k=100935 with 60774 digits, 607 digits more than Ray's.

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[Non-text portions of this message have been removed] - View SourceZak Seidov wrote:
> But PrimeQ works for n < 10^16 (?), Zak

I think you may have misinterpreted a passage in the documentation. It

notes the following:

* PrimeQ first tests for divisibility using small primes, then uses the

Miller-Rabin strong pseudoprime test base 2 and base 3, and then uses a Lucas

test.

* As of 1997, this procedure is known to be correct only for n < 10^16, and

it is conceivable that for larger it could claim a composite number to be prime.

This *doesn't* mean that PrimeQ doesn't work for numbers greater than

10^16, only that this test has been shown to be sufficient to *prove*

primality for all numbers less than 10^16. (The bound is probably higher

now... does anyone know?) Beyond that, it's not *proven* to always return the

right answer, but the probability is very high indeed that it will. The

probability of your hardware failing and returning the wrong answer is much,

much higher. In fact, there are *no* known counter-examples, as far as I know.

If you're familiar with the Rabin-Miller test, it's a probabilistic test.

It may incorrectly declare a composite number to be prime, but the probability

of error decreases with the number of bases that are tested. The probability

of error decreases quite rapidly as the numbers become large. See:

Damgård, I.; Landrock, P.; and Pomerance, C. "Average Case Error Estimates for

the Strong Probable Prime Test." Math. Comput. 61, 177-194, 1993.

However, this is then combined with a Lucas test. The combination of these

tests is often called the "Baillie-PSW" test.

Someone may correct me on this, but there are *no known* composite numbers

that are declared prime by this combination of tests, but it hasn't been

proven that there *can't be.* Carl Pomerance speculated that there might be

some, and gave hints on how said numbers could be constructed:

http://www.pseudoprime.com/dopo.pdf

It's a *very* strong test. Marcel Martin (author of the excellent PRIMO

prime-proving program) conjectured last month that "it is presumable that no

composite less than, say, 10000 digits can fool this test." See here:

http://tinyurl.com/3go6h

Several people have offers of money if you can find a counter-example. Not

a huge amount, but the combined totals will be easily over $1000.

--

Alan Eliasen | "You cannot reason a person out of a

eliasen@... | position he did not reason himself

http://futureboy.homeip.net/ | into in the first place."

| --Jonathan Swift

--

Alan Eliasen | "You cannot reason a person out of a

eliasen@... | position he did not reason himself

http://futureboy.homeip.net/ | into in the first place."

| --Jonathan Swift