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

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.