Prime numbers problem?

Question

Suppose I choose a prime Pn. Is the number P1*P2*...*Pn +1 always a prime?

here P1 = 2, P2 = 3 and so on. Pn is the nth prime number.
can anyone prove the primeness of the number?

What about P1*P2*P3...*Pn -1 ?

For one thing none of the primes 2,3,5,7,...Pn divide the two numbers. So they should be primes, right?

If they are indeed primes, don't we have infinite twin primes?

Someone please tell me where this argument is wrong .

Answer 1

Well it fails at P6:
2*3*5*7*11*13 = 30030
and 30031 = 59*509.
For P1...P17-1 = 510509
the factors are 61 and 8369.
You are right. None of 2,3,5,7, ..., Pn divide the
numbers but primes bigger than Pn do.
That's the essence of the argument
that there are infinitely many primes.
See the reference
http://mathworld.wolfram.com/EuclidNumber.html
for more information about these numbers,
called Euclid numbers.

Answer 2

Although the number Pn+1 is the one plus product of P1 through Pn, and is certainly not divisible by any of those primes, there are many other candidate primes which could divide Pn+1. In fact, at n grows, we know about a smaller and smaller fraction of the primes that could divide the Pn+1.

Consider, though, your case of P7, which is:
2*3*5*7*11*13*17 + 1 = 510511

Possible primes which could divide this number go up to 714.

In fact, the factors of 510511 are 19, 97, and 277.

Your conjecture is demonstrably false.

Answer 3

There could be a prime greater than Pn that divides this number .There will of course have to be a prime less than the square root of the number that will divide it so you only have to test as far as the sq rt.

Consider product 2*3*5*7 = 210

210-1 =209. 11 is a factor of 209. 11*19.

Answer 4

We need to re-examine what Euclid's proof there are infinitely many primes, given as Proposition 20 in Book IX of his Elements, actually says. I think it is being misremembered or misquoted here.

The oldest known proof for the statement that there are infinitely many prime numbers is given by the Greek mathematician Euclid in his Elements (Book IX, Proposition 20). Euclid states the result as "there are more than any given [finite] number of primes", and his proof is essentially the following:

Suppose you have a finite number of primes. Call this number m. Multiply all m primes together and add one. The resulting number is not divisible by any of the finite set of primes, because dividing by any of these would give a remainder of one.

Because all non-prime numbers can be decomposed into a product of underlying primes, this resultant number is either prime itself because it cannot be decomposed into primes, or there is a prime number or prime numbers which the resultant number could be decomposed into but are not in the original set of m primes.

Either way, there is at least an (m+1)st prime that was not accounted for. This argument applies no matter what m is; it applies to m + 1, too. So there are more primes than any given finite number.

This previous argument explains why the product of m primes plus 1 must be divisible by some prime not among the m primes, or be prime itself. A common mistake is to forget the first possibility, and think that Euclid's proof says the prime product plus 1 is always prime

It ain't, as the song says, necessarily so!

Answer 5

No, neither is always prime.

2*3*5*7 - 1 = 11*19

2*3*5*7*11*13 + 1 = 59*509

To answer the second part of your question, while it's true that these primorials are not divisible by any of the primes in their formulas, their size exceeds the square of their largest prime as soon as primes greater than 3 are involved, therefore the argument you mentioned fails.

Answer 6

the main important section you're becoming to be is 80 one sqft with a 9x9 sq. to get the 36 foot perimeter. despite the fact that 9 isn't top, so pass with the subsequent available numbers which will upload as much as a 36 foot perimeter, 7 and 11. So the section is 77 sqft. next is 13 and 5, yet that basically provides sixty 5 sqft, so the 7x11 rectangle is your answer.

Answer 7

P1*P2*...*Pn +1 is not going to be a prime number because prime numbers are only divisible by themselves and 1. The number X*Y*Z is going to be divisible by x, y, and z....and therefore not prime.