Analysis Questions

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February 4th 2008, 10:05 PM
heathrowjohnny

Analysis Questions

1. Prove that there is no largest prime.

Proof: Assume by contradiction that there is a largest prime $n$ . Then its divisors are $1$ and $n$ . Then how would I continue?

2. If $n$ is a positive integer, prove that the algebraic identity $a^{n} - b^{n} = (a-b) \sum_{0}^{n-1} a^{k} b^{n-1-k}$ . So use induction on $n$ ?

3. If $2^{n}-1$ is prime, prove that $n$ is prime. Then $2^{n}$ is not prime. But this does not imply that $n$ is prime.
February 4th 2008, 10:14 PM
Ryan J

for your first question: first imagine there is a largest prime number, X.

Think about how you could prove there is a larger one.

hint: think about factorials
February 4th 2008, 10:34 PM
earboth

Quote:

Originally Posted by heathrowjohnny

1. Prove that there is no largest prime.

Proof: Assume by contradiction that there is a largest prime $n$ . Then its divisors are $1$ and $n$ . Then how would I continue?

...

Have a look here: http://www.mathhelpforum.com/math-he...t+prime+number
February 5th 2008, 07:57 AM
ThePerfectHacker

Quote:

Originally Posted by heathrowjohnny

3. If $2^{n}-1$ is prime, prove that $n$ is prime. Then $2^{n}$ is not prime. But this does not imply that $n$ is prime.

Yes. If $n=pq$ where $p,q>1$ then $2^n - 1 = (2^q)^p - 1$ and use the factorization trick above. Which means it cannot be prime.
February 5th 2008, 09:08 AM
Jameson

Quote:

Originally Posted by earboth

Have a look here: http://www.mathhelpforum.com/math-he...t+prime+number

Good link.

OP - You're on the right track with assuming that there exists a largest prime and heading towards a contradiction. Call the largest prime $p_n$ . Take the product $p_{1}p_{2}...p_n$ . This is obviously not prime. What happens if you add 1 to the product? What divides $p_{1}p_{2}...p_n +1$ ?

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