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In that proof I apply that:

n=p!+1 (where p is a prime number)

this number (n) is not divisible with any prime number less than or equal to p. Why is that? Is there anyone who could please explain this to me or maybe make it probale to me?

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- Thread starter lonerider
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- #1

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In that proof I apply that:

n=p!+1 (where p is a prime number)

this number (n) is not divisible with any prime number less than or equal to p. Why is that? Is there anyone who could please explain this to me or maybe make it probale to me?

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The important lemma in infintely many primes is that every number is a product of primes. Suppose there are only finitely many primes. This guarantees the existence of a largest prime p. Consider the number n=p!+1. Now this number must be a product of primes (by the lemma). What can you say about all the other primes induced by the hypothesis? Remember p was the largest prime. Now what does this tell you about n? A very common mistake with students is that they will say n is prime. This is not necessarily true. Why?

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What can you say if a number d divides b+c? What can you say if a number d does not divide b+c? Suppose d divides b but does not divide c, does d divide b+c?

Edit: Okay, Oops. I misread you.

Every number must be a product of primes. That is, every number must be divisible by some prime.

In our proof, we supposed that there were finitely many of them. This means that there exists a maximal prime p. So now, consider n=p!+1. Can our primes divide n? Why or why not?

Edit: Okay, Oops. I misread you.

Every number must be a product of primes. That is, every number must be divisible by some prime.

In our proof, we supposed that there were finitely many of them. This means that there exists a maximal prime p. So now, consider n=p!+1. Can our primes divide n? Why or why not?

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no number and therefore it is a prime number :)

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no number and therefore it is a prime number :)

Be careful! It's not necessarily prime. Why?

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As p is the biggest prime number n should be divisble with some number. We do not know any number which is divisble with p! and 1 but 1. And we have to find a number larger than that - but we cannot. As that the definition of a prime number n must be a prime number and as it is bigger than p the statement has been proven false which proves that there is no biggest prime number which means there is an infinite number of prime numbers.

Am I on the right track?

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- #12

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- #13

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Do you mean that any number between 1 and p does not divide p!+1? Of course you don't have to prove it, but it's an easy proof. It builds 'mathematical maturity'. Besides, there's nothing I hate more than one of those damn combinatorial paragraph proofs.

- #14

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try to derive a contradiction by considering the following:

Assume that p is the largest prime number.

Either

(i) n is prime or

(ii) n is not prime.

Derive a contradiction.

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