## Comment Re:Open set it is! (Score 1) 248

Gah.

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Gah.

I did. I've also now tested it more fully and I think it's wrong.

The product of the first 75 primes (up to 379) is X=1719 620105 458406 433483 340568 317543 019584 575635 895742 560438 771105 058321 655238 562613 083979 651479 555788 009994 557822 024565 226932 906295 208262 756822 275663 694110

2^X mod X+1 is 1 so X+1 is probably not prime.

Tim.

No!

The original proof is flat out wrong.

It claimed that the product of all the primes up to N and then adding 1 is prime.

The product of all the primes up to 13 is 30031.

2^30030 mod 30031 is 21335

Therefore 30031 is not prime. (Fermat's little theorem)

The proof as stated is insufficient. I have proved that 30031 is not prime but I have not found any prime factors of 30031. Therefore, to complete the proof you need to include the case that 30031 contains a prime factor not in my original list because I have proved that 30031 does not belong in my list.

Tim.

http://www.math.psu.edu/sellersj/courses/math035/fa11/handouts/07_infinitely_many_primes.pdf

1) This proof is not a âoeconstructiveâ proof. We do not build an infinite list of primes in the process. This is a proof by contradiction.

Yes, Sorry. My comment wasn't very well written.

What I was trying to say is that given a purported complete set of primes, it's impossible to construct a prime not in the list other than by first assuming that there is a prime not in the list.

Any algorithm that tests N+1, N+2... will not terminate if N is the largest prime.

Tim.

Oops, sorry, I misread your algorithm.

But it still doesn't help. You've assumed that it will terminate (i.e. there is a prime larger than N). So you cannot use it to prove that there is a prime larger than N.

Tim.

N!+1 either is prime or has prime factors not in 1...N. Try factorizing the integers N+1

... N!+1 in turn until you come to one that is prime.

Why on earth would I do that?

PRIMES is in P. So if I want a prime bigger than N I'd start testing numbers of the form N+s for increasing s for primality until I found one. (Actually for very large values of N there are better candidates to test which have special case tests for primality that are particularly fast)

I wouldn't try to find a prime bigger than N by trying to factor a number greater than N! While the algorithmic complexity of factoring isn't known it's almost certainly not in P.

Tim.

It's as much my fault. I'll try not to do that again!

Have you considered the possibility that I might be replying to a particular comment in the parent my post was attached to?

The comment in particular that I was replying to had:

"but it's not true for every prime number, so there **can** still be increasingly large gaps."

To which my reply said:

"Not only **can** there be increasingly large gaps but there **are** increasingly large gaps."

Tim.

I don't see why it gets around this problem.

The equivalent claim would be that

N!+1 is prime.

The correct claim is that N!+1 is prime or is divisible by a prime larger than N

The faulty proofs are trying to construct a prime not in the set. The correct proofs are showing that a prime exists that is not in the set without making any claims about what that prime is other than it's bigger than N.

I'm pretty sure that it has been proved that there cannot be a constructive proof that there are an infinite number of primes - i.e. there is no way to construct a prime larger than N for arbitrary N.

Tim.

No! Why is this causing so much confusion.

I claim that SEO (Some enormous number) is the largest prime.

You construct 2*3*5*7*11*...*SEO+1 and claim that it is a prime not in my list.

I run a quick probabilistic primality test and prove that your number is composite. (which it almost certainly is)

Conjecture: There are no numbers of the form 2*3*...*P_{n-1}*P_n + 1 that are prime for P_n greater than 11.

Not only can there be increasingly large gaps but there are increasingly large gaps.

(N+1)!+2 to (N+1)!+N+1 are N consecutive composite numbers - divisible by 2..N+1 respectively.

Therefore there are arbitrarily long sequences of composite numbers.

Tim.

Your proof as written is wrong.

I claim there are a finite number of primes viz:

2 3 5 7 11 13.

You construct 2*3*5*7*11*13+1 = 30031 and claim that this is a new prime in my list.

I say - no it's not 30031 is composite. (59*509)

--

The correct proof is to say that X+1 is either prime or is divisible by a prime not in the list thus proving that the list is incomplete. If the list contains all the primes up to N then there must be a prime bigger than N.

Or the number is divisible by a prime that wasn't in you initial set.

GP has already used all the supposed finite number of prime numbers in constructing his contradictory bigger prime.

The GP's correction is right.

The GGP said that his number was prime. It might be, but it might not. But if it's composite then it cannot be divisible by any of the primes in his initial set so there must be a prime not in his set.

For example, if we assume 13 is the last prime then multiply them all together and add 1 we get 30031. But 30031 is not prime - it's divisible by 59 (which is a prime not in our set)

Tim.

*This is about minors, kids, who end up getting porn on there phones/tablets by accident, while looking for something innocent.*

I don't think it's anything to do with "accident". Yes, occasionally, some idiot posts some disguised link and managed to get people to click on it (I've been caught out a couple of times in slashdot posts before they started putting the domain after the link)

But mostly I suspect it's 13+ year olds going looking for porn. Which we all did. Back in my day though it was all still photography in magazines. VHS was around but I never even saw a pornographic VHS cassette, let alone watch what was on it.

One of the problems is that what is depicted in porn, especially today when video is easily accessible, is not "normal." Children are getting a distorted view of what normal relationships are about.

The solution, however, is to educate them about what normal relationships are. To teach them that porn is, for the most part, a fantasy. That within a normal sexual relationship some people do live out some fantasies and that it's OK for others to say "that's not for me" and that it's not OK to pressure others into doing things that they aren't willing to do or ready for. It's most especially not OK to use porn as an example to say "well they're doing it."

A porn block won't work. Teenage boys are going to get porn regardless of what adults try. Accept that fact and work with it instead of trying to deny it.

Tim.

Executive ability is deciding quickly and getting somebody else to do the work. -- John G. Pollard