Prove that there exists an infinitely large number of primes.

After some people see this elegant solution (I think Euler did it for the first time) of proving that there are infinitely many primes, they usually think that p1*p2*p3*...*pn + 1 (the product of the first n primes, plus one) is always prime. In fact, this is not true. The first counterexample is 2*3*5*7*11*13 + 1 = 30031 = 59*509.


Why does this happen?? Because we are not considering if p1*p2*p3*...*pn + 1 is divisible by a prime larger than pn.


It's also curious that there's no way of generating primes in an easy way (or in a short time)...

Posted by Fernando on 2003-03-25 14:42:16