# FOM: reply to Torkel on proof of infinity of primes

Torkel Franzen torkel at sm.luth.se
Wed Nov 26 04:40:41 EST 1997

```  Neil says:

>Theorem. For every prime p, there is some prime > p.

>First proof. Take an arbitrary prime p. Let k1,...,kn=p be all the
>primes less than or equal to p. Consider (k1*...*kn)+1 and apply the
>method of the proof of Lemma 1. [NOTE THAT IN THIS PROOF ONE *IS*
>USING THE PRIMALITY OF EACH ki IN ORDER TO CONCLUDE THAT ANY PRIME
>FACTOR OF (k1*...kn)+1 WOULD BE > p.]

The step from "q is a prime different from each of k1,..kn" to "q is
a prime greater than p" does indeed use the assumption that all primes
smaller than or equal to p are among k1,..kn. However, the assumption
that each ki is actually prime isn't used.  Hence a very fastidious
relevantist might want to weaken the assumption about k1,..kn.

Anyway, your "first proof" is a splendid one, but it is clearly
different from the common argument that I used in my illustration,
which begins by introducing the completely idle assumption that
p1,..pn are all the primes there are. No such assumption occurs in