Why might this be true? We list numbers of the form and underline those that are prime:
Not only is it plausible that underlined numbers will continue to appear indefinitely, it is something we can easily prove:
Then for any . Moreover, not every prime is of the form ; if they all were, then would be of the form . Thus there is a that is of the form . Since for any , we have found a new prime of the form . We can repeat this process indefinitely, so the set of primes of the form cannot be finite.
is a prime of the form . Next
which is again a prime of the form . Again:
This time is a prime, but it is of the form . However, is prime and . We are unstoppable:
This time the small prime, , is of the form and the large one is of the form .
William 2007-06-01