Question

Show that there are infinitely many positive primes ??

Answer 1

If possible, let there be finite number of positive primes p1,p2,,pnSuch that p1<p2<p3<...<pn.

Let x=1+p1p2p3...pn. Clearly p1p2pn is divisible by each of p1,p2,p3<...<pn.

∴ x=1+p1p2p3...pn is not divisible by any one of p1,p2,...,pn 

⇒ x is a prime or it has prime divisors other then p1,p2,...,pn

There exists a positive prime different from p1,p2,...,pn

This contradicts that there are finite number of positive primes.

Hence, the number of positive primes in infinite.

Answer 2

Answer:

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