# Unsolved problems

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 1414273.  Sun Jul 03, 2022 9:56 am Surely one of the more famous unsolved problems has to be Maria. PDR 1414275.  Sun Jul 03, 2022 10:14 am As for what makes it onto the show; if Godel's Incompleteness Theorem hasn't already been mentioned, then I reckon it's the most explicable and fun one. 1414276.  Sun Jul 03, 2022 10:17 am ConorOberstIsGo wrote: As for what makes it onto the show; if Godel's Incompleteness Theorem hasn't already been mentioned, then I reckon it's the most explicable and fun one.

Gödel's Incompleteness Theorem isn't an unsolved problem. It was proved by Kurt Gödel in 1931. 1414286.  Sun Jul 03, 2022 2:45 pm Quote: Surely one of the more famous unsolved problems has to be Maria.

See also: how to catch a cloud and pin it down. 1414293.  Sun Jul 03, 2022 5:45 pm Quote: Fermat's Last Theorem, I think you mean. Fermat's theorem (usually known as Fermat's little theorem to distinguish) states that if p is a prime number, then for any integer a, the number a^p − a is an integer multiple of p. The proof isn't too difficult (I remember doing it as an undergraduate).

Am I missing something here?

Let p=3 (a prime number as required), and a=2 (an integer)

Then a^p = 2^3 = 8

which is not an integer multiple of p (alias 3)

oh - and you're right - the specific theorem of Fermat that is commonly known as his last theorem rather than any of the other theorems with which it could have been misconstrued - apologies 1414294.  Sun Jul 03, 2022 5:52 pm Quote: "Fermats unwillingness to provide proofs for his assertions was all too common. Sometimes he really had a proof, other times not."

A statement is not evidence - what evidence do the authors of this statement advance to show that there were occasions when M Fermat claimed a proof while not actually having one to hand? 1414295.  Sun Jul 03, 2022 5:56 pm Incidentally - the paradoxical nature of time travel - I've solved that one and you'll hear about it yesterday 1414297.  Sun Jul 03, 2022 6:45 pm Brock wrote: Gödel's Incompleteness Theorem isn't an unsolved problem. It was proved by Kurt Gödel in 1931.

Apols Brock, I didn't mean it should be listed under 'unsolved problems'; I meant it should be a key thing to mention while talking about unsolved problems more broadly. Not sure what the unsolved problem should be because I'm not hugely enamoured with any mentioned so far.

Which one of the 23 unsolved problems remain of Hilbert's problems? They were pretty influential. 1414298.  Sun Jul 03, 2022 7:48 pm bobwilson wrote:
 Quote: Fermat's Last Theorem, I think you mean. Fermat's theorem (usually known as Fermat's little theorem to distinguish) states that if p is a prime number, then for any integer a, the number a^p − a is an integer multiple of p. The proof isn't too difficult (I remember doing it as an undergraduate).

Am I missing something here?

Let p=3 (a prime number as required), and a=2 (an integer)

Then a^p = 2^3 = 8

which is not an integer multiple of p (alias 3)

The Little Theorem is usually stated as: a^p is congruent to a(mod p)
(In simpler language (a^p) divided by p leaves a remainder of a.)

The version quoted: p divides (a^p - a), is exactly equivalent.

In your example, you forgot to subtract a from a^p.

2^3 - 2 = 6 1414308.  Mon Jul 04, 2022 2:40 am ConorOberstIsGo wrote: Which one of the 23 unsolved problems remain of Hilbert's problems? They were pretty influential.

There's a list here. I think they're all too advanced for a lay audience though. Page 2 of 2
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