A question regarding Fermat's Last Theorem?

Question

Could some genius please help me to explain this theorem? As in, give me some examples in layman's terms. Thank you.

Haha, I am not expecting the PROOF, so nobody needs to type 130 pages of Andrew Wiles' proof.

I kinda need examples... what do the three unknowns aka A B and C represent? For pythagoras theorem I know it represents the hypotenuse, breadth and height of the right-angled triangle, but did Pierre de Fermat have no purpose whatsoever for this theorem??

Answer 1

Let's start with the complex definition and break it up. Fermat's theorem states:

"if an integer n is greater than 2, then the equation a^n + b^n = c^n has no solutions in non-zero integers a, b, and c."

So, what it's saying is that for ANY three integers a, b, and c (not including 0) you wouldn't be able to raise a, b, and c to an integer, n, which is greater than 2 so that a^n + b^n = c^n.

If I was to let a = 6, b = 2, and c = 8, then I wouldn't be able to find an integer greater than 2 that I could raise all the numbers to and get: 6^n + 2^n = 8^n.

Answer 2

Fermat didn't have any geometric interpretation for a, b, and c that's analogous to the sides of a right triangle in the Pythagorean theorem. It was more that he was fascinated by the fact that one could satisfy the equation a^2 + b^2 = c^2 with *integer* solutions. For the Pythagoreans, integer solutions weren't so important. In fact, they were more interested in the non-integer solutions such as (1,1,sqrt(2)).

But Fermat (and I'm sure others at the time) were interested in the integer triples such as (3,4,5) and (5,12,13), and they wondered if they could find integer triples that satisfied similar equations, like a^3 + b^3 = c^3. but their curiosity here was completely unrelated to any geometric purpose.

as you may know, wiles' proof heavily utilized geometry, but the geometry he used is very modern and would have been completely unfamiliar to fermat.

Answer 3

Fermat's Last Theorem:
For any three whole numbers:
A^3 + B^3 not= C^3
A^4 + B^4 not= C^4
A^5 + B^5 not= C^5
... and so on.

I could explain it some more, but there isn't enough space in this little box ...

Answer 4

all of that's mindless in any appreciate, your 4th. variety question coated. Sorry to be so direct P.S. The sentence "x is the buyer-friendly term between a,b and c" isn't an many cases defined terminology, in case you pass with to persist with it you're in a position to prefer to define that, is it x=GCD(a,b,c) the only appropriate client-friendly Divisor? etc... make the hassle to be certain what a theorem or any mathematical fact particularly means. as quickly as you're saying, as an party: "all persons understand , a^(n)=c^(2)-b^(2)" you're in a position to prefer to define which a,b,c are, are they any 3 integers? or you propose "enable a,b,c are 3 integers verifing that equation"; or you're making purely confusion for your self and for others, that would now no longer sensible to you, particularly that would now no longer arithmetic. in case you particularly love arithmetic (as i visit hint out of your interest) take time to income the common mathematical language first, and the best judgment on the lower back of each and every and each math techniques. i think of of you're in a position to do sturdy themes with somewhat greater of humility and of direction with your interest ideal to expertise and clarity. thank you to your interest

Answer 5

Fermat's Last Theorem

a^x + b^x = c^x

where a, b, c, and x are integers

has no nonzero solutions for x>2.
,

Answer 6

Beat me to it . I was going to prove it ................but can't be bothered to type out hundreds of pages this evening.