Malteaser

Can anyone help us here? Anyone any views/comments?

For what it's worth Tony McC, I found your workings interesting. What's really tough with anything like this is that as soon as you start looking at other people's work then you get in to some really complicated Maths that you quite quickly can't understand. I have a degree in Maths and Stats and I can't follow the kind of explanations that are given on some websites and texts either. Perhaps it's a function of the human mind (or my mind!) that once someone starts explaining why an idea is not a goer then there is less of an inclilation to try and understand what they are saying! Having said that. Thanks for your thoughts. I would be interested in some of your other thoughts too, for what it's worth. Even if they don't lead anywhere, I'd still be interested. But we may have to put up with some acedemic/intellectual goading in the process! As for Uncool's work, I couldn't quite follow it and can also see merit in seeing the description in visual form. if this could be provided, that would be great. I think about this when i go to sleep, and don't get very far! My simple steps go along the lines of the following:- X^n + Y^n = Z^n rearrange to get : X^n = Z^n - Y^n which can, I think, always be factorised by (Z-Y) Hence, for example, using n=3: Y^3 = (Z^2 + ZY + Y^2)*(Z-Y). At this stage I then tried to work out what I can say about Z^2+ZY+Y^2. Firstly what doesn't it factorise by? Z? Y?, Z-Y?, Z+Y?, We can also draw up some rules about mutual exclusivity of X, Y and Z and potentially deduce that it won't factorise into 2 and 3. It gets more complicated with factorising where n>2 as higher powers of n mean my equation can have more factors than just (Z-Y), but I think you can always say that a part of the factorising has a larger power than 1 in it and this, therefore means we can draw similar conclusions as above on those parts of the equation What I would then be looking to do would be to come up with a set of things that we can say about X, potnetially then proving that X cannot have a set of factors and therefore limiting would X could be - and contradicting what we originally said that X could be. But evidently I'm far from that with the above. Any thoughts? Has this all been done and discarded before? As a further thought can we do the saem kind of deduction on X^n +Y^n. Can we say X isn't a factor? neither is Y? Neither is X+Y, Neither is X-Y? Neither therefore is 3 and by putting some exclusivity constraints on X and Y, neither is 2. Can we say this or can we say something similar and is it important that we can or can't say any of this?

All Given this statement, I've been wondering for some time now whether you can prove that (z-x)^2 is definitely not a divisor of z^n - x^n. I think I can prove it iff (if an only if) (z-x) is divisible by z^(n-1). But I'm not sure I can prove it for all z. Can anyone help? The proof for n=3 would be something like: x^3 = z^3 -3(z-x)x^2 -3(z-x)x - (z-x)^3. (you could prove this by drawing a cube of x^3 and working out which cuboids are needed to get z^3) Which means that z^3 - x^3 = 3(z-x)x^2 +3 (z-x)x + (z-x)^3. Evidently this is divisible a (z-x), but is only divisble by (z-x)^2 if x^2 is divisble by (z-x). Any help/pointers would be great. Thanks. Malteaser As an aside, what does non-incidental mean please? I'm guessing it means mutually exclusive?

dorcus I'm still not 100% sure we've taken the right approach myself - D H should be able to tell you that for sure. As for your question. You are right I can't divide each part of a function by different expressions, that would be wrong and could get you in trouble. What I can do is say that W^10 = W^3 * W^3 * W^3 * W. But I know from D H's clever method of multilpying through by x-1 that x^3 and therefore W^3 is equal to 1. So all I need to do is substitute each W^3 with 1. I'm not dividing by anything. What level Math is this by the way? Looks like UK A-level to me? It might be worth trying to work out the answer using complex numbers to prove our answer - but I'm told you might need to use trig at some point and I can't remeber trig very well. D H - any help here?

I might be wrong, but doesn't the cube root of 1 have three answers - 2 of which are complex? That leads you back to the solution of the original quadratic, which is also complex. The solution x=1 is the solution that we added when we multilied the original equation by x-1. But I'm still stumped by the next move at the moment. I guess you could say that since x^3 = 1 then you could divide 1 lot of x^3 from x^5 and 3 lots of x^3 from x^10. This leaves the equation w^2 + w + 3 = ? We know that x^2 + x = -1, from the original equation so substituting in we would end up with -1 + 3 = ? Which gives 2. How's that?

Was having a bit of fun with Fermat's last theorem and, when factorising x cubed minus y cubed, I came up with an interesting formula: x^2 - xy +y^2. I was looking at this formula to see if it would factorise, since the solution appears complex, I concluded that it didn't. However, I then realised that it's more important to solve x^2 - xy +y^2 + N. Which does have roots, I think, depending on the value of N. When I tried to solve it I came accross the function 3/4x^2, which can only be prime if x is an even number. i quickly realised this was a trivial statement. Bit it made me think that my original equation must be a prime if x and y are odd and mutually exclusive (though I'm not sure why I have to say the mutually exclusive bit). 1. Can someone help me to understand whether this is true and if not which numebrs x and y make it not true and why this is the case? 2. Also can some direct me to the polynomial theorem discussed above and whether this covers "polynomials" with an x and a y in them - are they still described as polynomials? 3. What is the actual question then re: prime numbers. I understood that the question was that there are no known formula that will always create a prime number. Is that the right question. Thanks Merged post follows: Consecutive posts mergedHaving done a simple spreadsheet my statement "it made me think that my original equation must be a prime if x and y are odd and mutually exclusive (though I'm not sure why I have to say the mutually exclusive bit)." is not true, But I'd still like to work out why it's not true if I can't find roots to my equation? and some tips on which way tio investigate this further?