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Brock
1414074.  Thu Jun 30, 2022 8:58 am Reply with quote

Is there a canonical example of a mathematical problem that's simple to state, but which no one has yet managed to prove? It always used to be Fermat's Last Theorem, until Andrew Wiles came along and proved it.

Goldbach's conjecture (every even number greater than 2 is the sum of two primes) is a strong contender, but I rather like the Collatz conjecture:

Start with any positive integer.
If it's even, divide it by 2. If it's odd, multiply by 3 and add 1.
Take the result and repeat the above line.

The conjecture is that the process will eventually reach 1, irrespective of which integer you start with. (After 1, it goes round in a loop: 1, 4, 2, 1 and so on.)

Example starting with 19 (odd values in bold):
19, 58, 29, 88, 44, 22, 11, 34, 17, 52, 26, 13, 40, 20, 10, 5, 16, 8, 4, 2, 1

This has been verified by computer for all starting values up to at least 2^68 (approximately 2.95 × 10^20). No one has yet found a starting value that doesn't reach 1, but there is as yet no proof that no such value exists.

Any other candidates?

 
suze
1414093.  Thu Jun 30, 2022 12:02 pm Reply with quote

The Parallel Postulate.

Draw a straight line - we shall call it A - and grant yourself the power to extend this line indefinitely in either direction. Then draw a dot - B - which does not lie on that straight line.

There is precisely one straight line which can be drawn through point B such that it never crosses line A.

That result is "obvious", and the line through B is of course parallel to line A. But the non-existence of further straight lines which never cross line A has never actually been proved. The whole of Euclidean Geometry (ie the geometry that we learned in school) is based on the assumption that the Parallel Postulate is valid.

 
CB27
1414095.  Thu Jun 30, 2022 12:10 pm Reply with quote

With regards Collatz, it's been a very important problem in creating models for pandemics given various data inputs. I seem to recall when Covid19 hit there was a model using Collatz's sequence, which was ignored by many western nations at the start, but was remarkably accurate in drawing predictions for infections and deaths.

 
Dix
1414101.  Thu Jun 30, 2022 12:39 pm Reply with quote

suze wrote:
The Parallel Postulate.

That's not a problem. There is nothing to solve.
The parallel postulate is one of the basic assumptions that Euclid set out, along with the definitions concepts of point, line, circle and right angle.

But it's not as straightforward as the other ones.

You can assume it to be true, and you'll get the classic Euclidean geometry. He wouldn't have got very far without it.

But you can also assume it not to be true in various different ways, and you get different but internally consistent geometries.

I quite like the hyperbolic one.

 
Dix
1414108.  Thu Jun 30, 2022 1:04 pm Reply with quote

Brock wrote:
Any other candidates?

Someone has helpfully listed the most well-known ones.

I'd say the Goldbach. It's simple to understand, it can be stated in one fairly simple sentence, and the only mathematical knowledge required is that you know what primes are.

Some of the others are a bit... technical.

 
Brock
1414110.  Thu Jun 30, 2022 1:47 pm Reply with quote

Dix wrote:
Brock wrote:
Any other candidates?

Someone has helpfully listed the most well-known ones.


Thanks for that. Out of those I'd say that the only "non-technical" ones are the Goldbach conjecture (1), the twin prime conjecture (4), the Collatz problem (6), the 196-algorithm (7), sums of cubic numbers (12) and proving whether odd perfect numbers exist (16).

Quote:
I'd say the Goldbach. It's simple to understand, it can be stated in one fairly simple sentence, and the only mathematical knowledge required is that you know what primes are.


Is it possible that the Goldbach conjecture simply cannot be either proved or disproved? I have long suspected that it's essentially a statistical result - the higher the even number, the more likely it is that you'll be able to find two primes that add up to it. There might be no number-theoretic reason for it.

 
Dix
1414111.  Thu Jun 30, 2022 1:58 pm Reply with quote

<< 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.

:-)

 
bobwilson
1414208.  Fri Jul 01, 2022 4:22 pm Reply with quote

I'd quite like to put Fermat's Theorem back in this list - I know it's been solved, but I also know that's not how Fermat solved it because the mathematical techniques and cross-overs involved weren't know about in his day.

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

 
suze
1414211.  Fri Jul 01, 2022 4:33 pm Reply with quote

Let us put Herr Doktor Gödel back in his box, and get William of Ockham out in his place.

Fermat didn't prove the result, and he was bullshitting when he claimed that he had. Doesn't this seem the most likely explanation?

 
bobwilson
1414213.  Fri Jul 01, 2022 4:47 pm Reply with quote

I had considered that possibility but I think it unlikely because:

1. He made the claim in a marginal note of his own notebook which was discovered after his death - so "claimed" perhaps gives a misleading impression
2. 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)
3. It is entirely in keeping with his known character for him to leave such a statement about a proof which he knew would infuriate his rivals, and as far as I can recall, there are no known false or incorrect claims made by Fermat

So, whilst Mr Ockham may have a say, I believe there is some rationale for doubting the integrity of his testimony

 
Brock
1414225.  Sat Jul 02, 2022 2:34 am Reply with quote

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.

 
tetsabb
1414229.  Sat Jul 02, 2022 3:57 am Reply with quote

Errrr.... surely Fermat's last theorem or whatever concerned numbers to the power of x and Pythagoras?
While a˛+b˛=c˛, this does not work for any other number.

 
Brock
1414232.  Sat Jul 02, 2022 4:10 am Reply with quote

tetsabb wrote:
Errrr.... surely Fermat's last theorem or whatever concerned numbers to the power of x and Pythagoras?

While a˛+b˛=c˛, 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.

 
ali
1414260.  Sat Jul 02, 2022 4:20 pm Reply with quote

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).

 
suze
1414270.  Sun Jul 03, 2022 8:39 am Reply with quote

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.

 

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