Euler's Sad Alt*Columbo voice* Well, that all sounds good. A finite number of primes. That'll look good on the report. Uh, just one more thing, sir, before I go. What if you multiplied all of them together and added one? What would that be divisible by, sir?
intermobility_berlin 🌻🇪🇺@eulersposter a finite number of primes could be {2, 3}. 2 x 3 + 1 = 7, which is prime itself. 🤷
SibrosanMaybe I understand incorrectly what you mean, but if that were the set of primes Columbo was talking about, your observation would have proven his point.
@sibrosan @eulersposter what I mean is that any products of any finite number of prime numbers are odd (so that when you add 1 they become even, i.e., divisible by 2). But 2 is also a prime, and if 2 is part of the finite number of primes, then this idea doesn’t work any more. What I mean is that to me it seems that there cannot be any general assertion that applies equally to all finite sets of prime numbers.
I think you're not being consistent, because any products of any finite number of prime numbers would be even, not odd, if 2 were included.
So would it have worked if Columbo had said:
"What if you multiplied all of them, except 2, together and added one?"
Either that, or you could just suppose that Columbo, observant as ever, had already noticed that 2 was not included in the set.
@sibrosan „… any products of any finite number of prime numbers would be even, not odd, if 2 were included.“ Exactly my point!
I’m not sure I correctly understand the original question. Are we trying to come up with an assertion that equally holds for all possible finite sets of prime numbers? Are we trying to figure out the particular set of prime numbers Columbo is looking at? I’m lost. I hope @eulersposter can clarify.
"I’m not sure I correctly understand the original question"
Are you familiar with the detective series Columbo?
Its protagonist, portrayed by Peter Falk, is an apparently obtuse homicide department criminal investigator who, by posing seemingly pointless and innocent questions, at the end of each episode unmasks the murderer, who overestimated their own cleverness in evading the law.
@sibrosan @eulersposter I am. And here’s the explanation of the original post. It‘s not a riddle (as I assumed), but a classic mathematical proof that there are infinitely many prime numbers. Thanks to my friends who know more about math than I do, for explaining it to me. …
„Assume that there is a finite number of prime numbers. We can, therefore, list them as follows:
(p₁),(p₂),(p₃),…,(pₙ)
Now consider the number:
P=(p₁ ⋅ p₂ ⋅ p₃ ⋅ …⋅ pₙ)+1
We Notice that when you divide P by any of the primes listed above, the remainder will be 1. As a result, either P is prime, or its prime factorization contains only primes that were not on our list.“
Source: https://medium.com/math-simplified/proof-that-there-are-infinitely-many-primes-92516d139704
