GPT-4's proof of the infinitude of primes, as quoted in the paper, is wrong.
Yes, I think I can,
Though it might take a clever plan.
I’ll start by noting Euclid’s proof,
Which shows that primes aren’t just aloof.
Assume we have a finite list,
Of primes, and that none have been missed.
Multiply them all together,
And add one, just to be clever.
The result will be a number,
That has no prime factors, oh wonder!
Not true. The result may itself be prime. For example, say your finite list of primes is {2,3}. Multiplying them together and adding one yields 2*3 + 1 = 6 + 1 = 7, which is itself prime. Hence it has a prime factor: 7.
But every number has a prime,
That divides it, it’s just a matter of time.
So we’ve found a contradiction,
And our finite list needs eviction.
There must be infinitely many primes,
And that’s the end of my rhyming lines.
A correct verison of the proof might say that the result has a prime factor, however, that prime factor is not an element of the supposed finite list, hence the list is incomplete.
