Prime number

[kezman]

Hi i Have difficulties with this problem

Prove that exists infinite prime numbers.

Knowing that if there were finite prime numbers p1,p2,p3...pn then:

\(\displaystyle a = 1 + \prod\limits_{i = 1}^n {p_i }\)



is an integer different of 1 and -1 and cant be divided by any prime number.

 

[TchrWill]

Hi i Have difficulties with this problem

Prove that exists infinite prime numbers

There are an infinite number of Primes
Proof:
From Euclid's Elements (Proposition 20, Book IX)
Assume that p1, p2, p3, ......pn is a finite set of prime numbers.
Create the product P = p1(p2)p3.......(pn) and add 1.
P + 1 forms a new number which is not divisible by any of the given set of primes and must therefore itself be a prime or it contains as a factor a prime differing from those already defined.
If P + 1 is prime, then it is clearly a new prime not of the original set of primes.
If P + 1 is not prime, it must be divisible by some prime q.
However, q cannot be identical to any of the given prime numbers, p1, p2, p3,......pn, as it would then divide both P and P + 1 and consequently, their difference = 1.
However, q must be at least 2 and cannot therefore divide evenly into 1.
Therefore, q, being different from all the given primes, must be a new prime.

Caution: Care must be taken to realize that p1(p2)p3.......(pn) + 1 will not always produce a new prime and if it does, it is not necessarily the next prime.
Examples:
2x3 + 1 = 7 = a prime
2x3x5 + 1 = 31 = a prime
2x3x5x + 1 = 211 = a prime
2x3x5x7x11 + 1 = 2311 = a prime
2x3x5x7x11x13 + 1 = 30,031 = 59x509

 

[kezman]

Im sorry I dont understand clearly this part.

P + 1 forms a new number which is not divisible by any of the given set of primes and must therefore itself be a prime or it contains as a factor a prime differing from those already defined.

Maybe its easy but I cant see why:
\(\displaystyle \frac{{P + 1}}
{P} \neq q \in {\rm Z}\)

 

[Gene]

When you divide P by any of the primes (which you claimed were all the primes that exist) it will have a remainder of 1 so P must be a prime bigger than what you claimed was the biggest or have a prime factor bigger than what you claimed was the biggest.

 

[kezman]

Ok thanks for the help