Is there any other proof by which I can show that there is no largest prime?
I saw an example where it is proved with contradiction.(Idea is basically that of Euclid's proof)
Imagine that the largest prime prime is $13$.So, total number of primes we know are-$2,3,5,7,11,13$.
Now,if I do $(2\times3\times5\times7\times11\times13)+1=30031$.So, we can see that $30031$ is not divisible by $2,3,5,7,11,13$ as they leave remainder $1$. Also,as it is formed by multiplying only primes it does not have any other composite factors.We also see that $30031=59\times 509$.Which are again two primes.Thus,$13$ is not the largest prime.
What are the other ways to prove that there is no largest prime?
Thanks for any proof!!