a sixth-grade student argues that there are infinitely many primes because "there is no end to numbers." how do you respond?

Question

tkhunny · Answer

Why does it have to be a sixth grade student?

mathstudent55 · Answer

Unless you're a math genius, agree with him because it makes sense.

tkhunny · Answer

Even if you are a math genius, agree with him, since it is easily proven.

anonymous · Answer

How is it easily proven?

tkhunny · Answer

The standard proof is this...

Take all the primes that exist and write them down.  $p_{1}, p_{2}, p_{3},...,p_{n}$  It's a finite list.

There are no more!  That's all the primes that exist.

What if we multiply them all together?  $p_{1}\cdot p_{2}\cdot p_{3}...p_{n}$.

That's a REALLY big number.  Something funny about it, it is divisible by EVERY prime number.  That's cool.

What if we add 1 to it? $p_{1}\cdot p_{2}\cdot p_{3}...p_{n} + 1$

Well, that's an even bigger number and it is divisible by no listed prime.

Since we listed all the primes, or so we thought, we must have missed one.  Oops!

No end to primes.