Discovery on prime numbers?

[abskebabs]

I'm currently revising for a resit of my Impact of Maths module which I took this year and I found really hard, especially for a module outside my main discipline. This could explain a few of my recent posts:-p . But I have just been doing a question from the book which I am using to revise for half of the course, called Mathematical Puzzling by Tony Gardiner.

 

The question was: "3 and 5 differ by 2 and are both prime numbers. What is the next such pair? How many such pairs are there like this?"

 

Interestingly the 1st thing I thought of was 11 and 13:doh: . I kept looking for pairs though, and it didn't seem like there was necessarily a limit. I looked in the answers, and it said that it has been conjectured that there were an infinite No of such pairs but this has not been proved.

 

Now I already know that a product of all "known" prime numbers added to 1 produces a prime, and this constitutes the proof that there is no largest prime number. I had been pondering this before, but isn't it true that you could take the same product of all "known" prime Nos and subtract 1 to produce another prime? Because the difference is again only by 1, the nearest prime factor of 2 would still not be a factor of the resultant No, and hence it would still be prime, wouldn't it?

 

If this is true as well, then(deep breath); Doesn't it constitute a proof that there has to be at least an infinite No of such pairs because both of the above described formulas for generating primes could produce an infinite No, and for the same number of prime numbers could produce an infinite number of pairs that are separated by 2.

 

The conclusion seems so trivial, that I suspect I have been naive here and made a mistake. Or perhaps, the conclusion is not really so out of this world, I don't know. I would be very grateful if someone more mathematically advanced could enlighten me on the validity of what I have so far stated on this thread.

[timo]

Ok, I have to think about it some more, but what I can already say: You might have misunderstood the proof of the infinite number of primes. It's a proof by contradiction, similar to this (not being a mathematician, I think it's necessary to add the "similar to" part - I don't exactly know all the subtleties involved):

1) Assume there is a finite number of all primes (it doesn't really matter if they are "known" or not). Call the set p.

2) Then, the product P of all those primes exist.

3) P+1 does not divide by any of those primes. Hence either

3a) P must be prime.

3b) Or, there must be some missing prime P' that divides P.

4) If a prime P+1 is not in p (which seems obvious, dunno what the exact reason is), or a P' exists, then you have a contradiction to the assumption.

5) To clearify: This then means that there is no finite set of all primes and that therefore P+1 does not exist.

 

Ok, what do we learn from that:

- There is a non-finite number of primes. Hence, you cannot contruct new primes in the exact way you tried in the proof.

- Your next-best guess would be that you just take any finite set of primes and try to pull off that trick. Of course, you quickly find a set of test-primes where this won't work (namely at least all that don't contain 2).

- Next-best guess is taking the N lowest primes. This might work, but I don't really see a reason why the N+1'th prime shouldn't divide that product+1 (or -1). Might be interesting to plug that into the computer, it might spit out an example quite quickly.

 

EDIT: Product of 6 first primes plus 1: 2*3*5*7*11*13 +1 = 30031. 30031/59 = 509.

[abskebabs]

Thanks for the response, I think I understand now. The example you gave perfectly illustrates your point. 30031, would be considered a prime number if 2, 3, 5, 7, 11 and 13 accounted for all prime numbers. But they are not, and so the number could be prime(like I assumed), or could be divisible by another yet unknown prime, as you mentioned(in this case it was 59). The same seems obviously true if we try to suppose the same for the set P of all primes, as you have already mentioned(sorry to be a parrot).

 

I can see why the "formula"(or its alternative version) cannot be used to generate primes, as it is simply for the purposes of showing that there cannot be a finite set of primes in the proof of contradiction. It's been educational:-)

[Country Boy]

Thanks for the response, I think I understand now. The example you gave perfectly illustrates your point. 30031, would be considered a prime number if 2, 3, 5, 7, 11 and 13 accounted for all prime numbers.

Well, no, if 2, 3, 5, 7, 11, 13 "accounted for all prime numbers" then 30031 couldnt be a prime number since it is not one of those!

 

The point of Euclid's proof was that, assuming a finite number of primes, multiplying them all together and adding 1 gives a number not divisible by any of those primes (you always get a remainder of 1). EITHER that new number is prime or it is divisible by a prime number not in the list. In either case, you have a contradiction.

 

But they are not

You couldn't use this as a proof of that there are an infinite number of primes because you have now assumed the conclusion!

 

and so the number could be prime(like I assumed), or could be divisible by another yet unknown prime, as you mentioned(in this case it was 59). The same seems obviously true if we try to suppose the same for the set P of all primes, as you have already mentioned(sorry to be a parrot).

 

I can see why the "formula"(or its alternative version) cannot be used to generate primes, as it is simply for the purposes of showing that there cannot be a finite set of primes in the proof of contradiction. It's been educational:-)

 

I'm really just objecting to your specific wording. I think you have the correct idea.