[aquafox]

Not a paper, but

2×3×5×7×11×13 + 1 = 59×509

is a short counter example to the widespread misconception that adding one to the product of the first n consecutive prime numbers always yields a prime number.

The reason you get away with this in the infinitely-many-prime-numbers proof is that the new number may not be prime, but can be written as a product of primes that are distinct from the first n primes. Thus you still generate new prime numbers with this technique.

[tgv]

I think this is not a problem because 59 nor 509 are in the list of prime numbers used (on the left side). Euclid's proof merely states that for every list of prime numbers, there's a new one, not in the list. https://en.wikipedia.org/wiki/Euclid%27s_theorem#Euclid's_pr...

[Sniffnoy]

Yes, that's the point -- Euclid's proof doesn't require that, it works like you said; but it's a widespread misconception that it does work that way. It's a counterexample to the misconception, not to the real proof.

[_v7gu]

Under the assumptions of the contradicting statement, 59x509 is prime. You can’t factorize a number iff there are no prime divisors, and the right hand definition for prime numbers is much simpler for the purposes of the proof; no extra “prime” generation necessary.

[red_trumpet]

You are right, but I wouldn't emphasize this in the proof, for the following reasons:

1. Since you assume a contradictory statement, you can actually derive everything you want. So saying "it's true in that context" is pretty meaningless.

2. I think it adds an unnecessary step in the proof. "This new number is not divisible by any prime, therefore it is prime, contradiction as it is not in the list" compared to "It is not divisible by any prime, contradiction since any number is divisible by a prime". I think that is confusing.

3. For didactical reasons. It can leave the reader/student with the wrong impression that multiplying the first n primes and adding one always creates a new prime.

[blauditore]

Is it really a "widespread misconception"? Anyone who spends just a little time on this will notice, even without a strong mathematics background.

[thaumasiotes]

There are a gigantic number of widespread misconceptions that anyone who spent even a little time on the issue would immediately notice are false.

Phil Plait used to complain about people who told him the moon wasn't visible during the day.