Re: Analysis of Flibble’s Latest: Detecting vs. Simulating Infinite Recursion ZFC

On 21/05/2025 19:42, Keith Thompson wrote:

Richard Heathfield <rjh@cpax.org.uk> writes:
[...]

Definition: a prime number is an integer >= 2 with no divisors >= 2
except itself.
>
Hypothesis: there is no largest prime number.
>
Proof:
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Assume that a largest prime number Pn exists.
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Itemise all the prime numbers from P1(=2) up to Pn :
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P1 P2 P3 ... Pn
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Insert x symbols all the way along.
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P1 x P2 x P3 ... x Pn
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Add 1.
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The number thus calculated is not divisible by any prime in our list
(there being a remainder of 1 in each case), so the number calculated
is (a) prime, and (b) larger than Pn. Thus Pn is not the largest
prime. This contradicts the assumption made at the beginning, which
must therefore be false. Proof by contradiction.

 You've made a mistake in an otherwise valid proof.

Yes, I have.

The number
(P1 x P2 x P3 ... x Pn) + 1 *either* is prime *or* is composite
and a multiple of some prime number larger than Pn.
 For example, (2*3*5*7*11*13)+1 is 30031, which is 59*509.

...which I could have found myself if I'd only thought to look. I could even have done it with short ints. ;-)
--
Richard Heathfield
Email: rjh at cpax dot org dot uk
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