# Unsolved problems

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1414225.  Sat Jul 02, 2022 2:34 am bobwilson wrote: I'd quite like to put Fermat's Theorem back in this list

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).

 Quote: So how did FERMAT do it? That would be interesting to know.

He almost certainly didn't. He was in the habit of announcing results without a proof, and really the term "theorem" is a misnomer (or was until 1994); unproved results are normally known as "conjectures". Nor was it the last such result that he came up with; it was the last one remaining after proofs had been established for all the others!

It's possible, of course, that he thought he had a proof, but the proof contained an error. However no details have ever been found.

 Quote: It is one of very many other similar marginal notes also discovered after his death - all of which, if I recall correctly, have been proved correct (some very quickly, others took longer)

True, but it doesn't therefore follow that Fermat had his own proofs of them all. 1414229.  Sat Jul 02, 2022 3:57 am Errrr.... surely Fermat's last theorem or whatever concerned numbers to the power of x and Pythagoras? While a²+b²=c², this does not work for any other number. 1414232.  Sat Jul 02, 2022 4:10 am tetsabb wrote: Errrr.... surely Fermat's last theorem or whatever concerned numbers to the power of x and Pythagoras? While a²+b²=c², this does not work for any other number.

Yes, that's correct. Fermat's Last Theorem states that no three positive integers a, b, and c satisfy the equation a^n + b^n = c^n for any integer value of n greater than 2.

Fermat's (Little) Theorem is something else, as I said above. 1414260.  Sat Jul 02, 2022 4:20 pm Dix wrote: << enter mr Gödel >> Short answer: Yes, that is possible. There are conjectures that are true but not provable in any sufficiently advanced formal system. I expect that it's even harder to prove whether a conjecture is provable or not. But I can't prove it. :-)

It gets worse: if you can show that a statement is undecidable (i.e. no proof (or disproof) exists), then that proves the statement true - because if it is undecidable, then no counterexample can exist (because a counterexample is a disproof). 1414270.  Sun Jul 03, 2022 8:39 am Brock wrote: 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).

Just as an idle Sunday afternoon challenge, I asked the good husband if he could state Fermat's Little Theorem. He actually stated it in a slighly different but equivalent form, and he too remembers proving it while at university.

He then pulled a textbook from the shelf. This particular textbook is intended for the IB equivalent of Further Math*, and it includes that proof. It also mentions that Pierre de Fermat published little mathematical work himself - he made his living as a lawyer and also wrote poetry, but he was not a Professor of Mathematics - and includes the statement:

"Fermats unwillingness to provide proofs for his assertions was all too common. Sometimes he really had a proof, other times not."

* Blythe, P et al (2005). Mathematics for the international student: Mathematics HL (Options). Adelaide, Haese & Harris. 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 1 of 2
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