On: Do Infinite Pencils Exist?

[eesmith]

jwilk gave a counter-example which demonstrated my proof was incorrect (or at least incomplete).

[jimhefferon → eesmith]

You are mistaken. Your proof is correct. You assert that no prime less than or equal to N divides N!+1. The argument is clear. The other poster gives an example that does not gainsay that assertion.

From your assertion the contradiction follows that if N were to be the largest prime ...

[eesmith → jimhefferon]

I wrote "Construct N!+1. That is not divisible by 2..N, so it is prime".

That is not correct. 5!+1 = 121 which is not prime, as it is 11*11.

The correct statement would have been "so it is prime, or a composite with no prime factor <= N."

[jimhefferon → eesmith]

You left out your "a largest prime." If N were indeed the largest prime then there would be no larger prime to divide into N!+1 and make it composite.

In any event it is a long way from wrong, IMHO. :-)

[eesmith → jimhefferon]

If my construction doesn't work for all arbitrary primes then it won't be guaranteed to work for "a largest prime", which means it doesn't provide an example of one way "infinite" is treated in modern mathematics.

(Remember, the original essay described a constructive requirement for "infinite" - “If you think there are infinitely many, write them all down.” I wanted to demonstrate how to think about infinity without that requirement.)

If someone in New Zealand sends me a letter and it's delivered to neighbor, it's still delivered to the wrong house - even if it gets to me eventually.