fermat's theorem

[boswellnimrod]

Fermat said no positive integers fit the pattern a^n+b^n=c^n when N is greater than 2.
I assume someone realized this before but I just noticed that N can be greater than 2 if the number of elements on one side of the equation is equal to N. In other words for example if N=3, there must be three variables on one side a^n+b^n+c^n=d^n
and this works 3^3+4^3+5^3=6^3.
So it is simple geometry that is being described. For every value of N you need N variables raised to that power to make the equation work.
Any mathematicians out there that have seen this? I assume it is known but I just noticed this.

 

[Master_Ghost_Knight]

I personally never found the problem interesting enough for me to dedicate the time it deserves to ponder about it.
Fermat's theorem only applies to additions of 2 elements, nobody contests that you can do it for powers greater than 2 by adding more than 2 elements because it is common experience that you can.
Now that to say that for an addition of elements of power n you must have at least n elements in order for the sum to be a integer of power n, is something slightly different, it would be a generalization of the Fermat theorem.
Formally it could look like this:
For all n positive integer bigger or equal to 2, there doesn't exist a sequence S_i of positive integers such that SUM(i=[1;j]; S_i^n)=S_(j+1)^n implicates that j<n

Is this true? I don't know, it probably isn't or else more attention would have been payed to it (you are not the first person to think about this, I guarantee you that).
Unfortunately your are going to get out of here quite disappointed, I noticed that you used the sentence:

Any mathematicians out there that have seen this? I assume it is known but I just noticed this.

Which is not very mathematician thing to say. Mathematical statements have to be proven, the fact that "you think you have noticed a pattern" or "that you tried a thing for a while and it worked every time you tested it so far" is irrelevant, all it means is that you have not found an exception so far. You can try to do it for a million years and fail, but that isn't proof that a cleverer person couldn't come along and do it instead. In fact very often when you are doing mathematics you will come across similarly convincing patterns or things that appear to work with everything we tend to try first that with a more in depth study about it turns out not to be true.
In other words, the fact that you used the word "just noticed", tells me that you don't have proof and you can not be sure of it either because if you had you would say:
"I have just been able to prove this, does anyone know of a mathematician who has done this as well?"
(which would have been quite a feat). And unless it is proven true (or required to be proven true for some other interesting problem) it is not that much interesting.

It also tells me that your are not sufficiently versed in mathematics to actually be able to prove it yourself, because the more you learn about mathematics the more reserved you would be about making such sweeping mathematical statements (specially without proofs), you must have proof that it is true before you can say that it is true. If it was rather "just an interesting pattern that seams true but you haven't be able to come up with a proof yet (which has a name and it is called a conjecture)" and "you were just trying to find out more about work that other people done on it", then you would say something more like this:
"I have this conjecture which seams quite interesting, does anyone know of a mathematician who has noticed this as well or has possibly proven it?"

However, do not get disappointed about this and I encourage you not to stop thinking about the problem, when I was younger I used to do this sort of thing you are doing now (noticing patterns that seamed to work but couldn't prove or didn't even had the mathematical bases that would allow me to know for sure). Later when I had gain more knowledge I have revisited those problems, somethings I had proven not to be true and other things I managed to prove to be true (yet known for along time) (and there was one thing that was true, I have managed to prove it was true, that was proven by someone else to be true before I did but hasn't done so before the first time I taught about the problem). What it matters here and now is that you are interested in mathematics and thinking in this sort of thing gives you the kicks, this sort of thing helps you practice the ability of thinking (which does need allot of practice to do it right). And who knows later in life when you have a University degree you can revisit the problem with your new found knowledge and make some breakthrough.

And as a suggestion if you are truly interested, there is something that you can do to try and disprove it by means of an example. You can design a computer program that tries every possible combination of additions of the 4th power of 3 numbers between 1 and 100 (doing more than this will probably take a couple of days) and see if you can find one which that works (a search algorithm).
If you find one combination that works then you have disproved the conjecture (because the conjecture says that there isn't any, but because you have found one you can say that there is at least the one you have found therefore the conjecture can not be true). If you tried to find and failed, it doesn't prove that your conjecture is true (it just proves that you haven't found it) however it maybe a good indication that it might be true and that therefore you are justified in continuing using your time to think about it.
If you don't know how to do a program I can help you with that.

 

[boswellnimrod]

Hey thanks for the reply. Seeing the concept written out formally was interesting.
Yes I am not a mathematician. And unfortunately I do have a degree (not in that field), so I won't be pursuing that when I get older (I am older).
I am aware that this was more in the realm of an observation and to be of any value it would require a formal proof.

I was just wondering if a different approach had been tried for Fermat's last theorem, since he implied that the proof was fairly small. I thought (naively perhaps) that the types of proofs used in geometry might be used here and be less involved than the one that Andrew Wiles presented.
Separately, I think mathematics is the song of the universe and can reveal truths that would otherwise elude us. I think it is one of the greatest achievements of mankind.

 

[boswellnimrod]

I did a little more reading on this (which I suspect I should have done first). As far as I can make out, using the modularity theorem to solve Fermat was using geometry but at a far more complex level than I could ever understand.

 

[Master_Ghost_Knight]

Hey thanks for the reply. Seeing the concept written out formally was interesting.
Yes I am not a mathematician. And unfortunately I do have a degree (not in that field), so I won't be pursuing that when I get older (I am older).
I am aware that this was more in the realm of an observation and to be of any value it would require a formal proof.

I was just wondering if a different approach had been tried for Fermat's last theorem, since he implied that the proof was fairly small. I thought (naively perhaps) that the types of proofs used in geometry might be used here and be less involved than the one that Andrew Wiles presented.
Separately, I think mathematics is the song of the universe and can reveal truths that would otherwise elude us. I think it is one of the greatest achievements of mankind.

Unless you are 90 and with Alzheimer's, it is never to old to learn mathematics. And neither do I have a degree in mathematics, I'm an engineer (granted from a top University know for its outsanding mathematical expertise), the point is that didn't stoped me to know allot about this things even if not working directly in the field. Granted it is not easy to do it on your own, but there are good books out there. Mathematics has everything else in life requiers practice, and it is generaly a good idea to start by prooving simpler statments and work your way into more complicated ones. Very often mathematical problems can look deceptively simple and yet be extremely complicated to prove, on the other hand when you gain experience in handling this sort of object extremely complicated problems become in time easier to solve. People are not born knowing this things, even Einstein had it's days when he couldn't add 2 numbers togheter even to save his own life, sure he got where he got because he was smart but I would say that by far the greatest contribution was due to the training he had.
You have an interest in math, and I sugest not let it die.

As for Fermat's last theorem, many great mathematicians had worked on it for years and tryed thousands of different aproaches to no avail (and things that most people wouldn't even think about), of course you will hear of the one solution that happened to work but you will never hear of the thousands of other things that pople tried but didn't work (which happens to be about 99% of cases) .
There maybe a simpler proof to Fermat's theorem, after all Fermat said he could do it and it turnout to be that his conjecture was also right (which didn't had to be), however nobody knows what it was, he probably might have used a clever trick that nobody managed to think about. However it is all possible that Fermat didn't had no proof at all even tough he taught he did (and just happened to be lucky the theorem wa true after all but for different reasons), very often it happens to me (and it is also acommon experience for other mathematicians) that I have an idea to solve a problem and it seams very convincing that is going to take it home and that it works perfectly but when you write it down with every detail it turns out that it doesn't work and it becomes a dead end. There are many stories of mathematicians publishing work saying that they have managed to solve a problem, but then they later retract it saying that their poof actually didn't worked, and later they say they have managed to do it this time just to retract it again (and keep going like this back and forth for a long time).

 

[Pulsar]

Hello boswellnimrod,

your idea is very similar to Euler's sum of powers conjecture, which has been disproven.

cheers