Prime Numbers

micromass

OK, take 11, square root rounded down is 3. But 11/3! is not 11, so it doesn't remain unchanged...

PrimeNumbers

In this case the number as a whole changes from 11 to 11/3! but 11 itself doesn't change at all. If it were 25 and I divided it by 120 it would change to 5 and not remain the same. Basically I can write a fraction of X/(Square root of X rounded down)! on a piece of paper or calculator and change it to smaller numbers on both sides of the division line by either myself or the calculator if it is not prime.

atomthick

I believe PrimeNumbers wants to say that if [tex]GCD(N,[\sqrt{N}]!) = 1[/tex] then N is prime.

gb7nash

This is true, and can be proved using prime decomposition. Is it practical though? I'm not sure. If you want to determine if a humungous number is prime, calculating factorials and then the gcd can be a very excrutiating process.

atomthick

Clearly it's not computational feasible for large numbers, however there are some interesting results for example if [tex]gcd(N, [\sqrt{N}]!) = P, P > 1[/tex] then P is a factor of N.

It could become computational feasible if someone finds good algorithms for adding, subtracting and finding modulus that work in the factorial base (Cantor discovered that we can write any number in factorial base, for example 15 = 1! + 1*2! + 2*3!). Because we can easily find the representation of N in factorial base all we would need is fast computational algorithms for this base.

P.S. EIRE2003 see how many interesting questions are prime numbers raising? This kind of questions and their answers have made great improvements allover mathematics! Those numbers look uninteresting until you ask a question about them, try it.

Char. Limit

Huh? Take off the caps lock and size changes, please. And explain better. Have a comma: ","

epenguin

It was surely not for any of the applications that have been mentioned. It was cultivated for centuries before they were dreamt of. And also prime numbers specifically are not really essentially connected with cryptography. It is just that factorisation into prime numbers is one, just one, example of a hard (computationally very long) problem whose inverse (multiplying the factors) is not hard, if I understand. There are other such hard problems ready to take over for cryptography if ever anyone cracks the factorisation problem.

I think of it as having a pile of pebbles, can I arrange them in a regularly spaced rectangle? If not I have a prime number of pebbles. Could be tempted to wonder if it is worthy of a grown man's attention. Tempted to believe that it would be if it were simple - could be explained, followed, carried in the head it would be revealing of a structure. But if it is so difficult and complicated that no one understands the solution when it is found, will it be revealing in the same way? I believe this question is discussed about some of today's very difficult proofs. Unless it throws light on other problems whose significance is more apparent. We are told this would be so, but I suggest we do need to be told.

chhitiz

didn't ken ono recently establish something like 'factorials of primes follow a fractal pattern'?
can someone post the proof please?