Why two given equations cannot both be true (fermat's for n=3)

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In summary, the conversation is discussing the difficulty of proving Fermat's last theorem for n=3 and trying to understand why (x+y) cannot equal a cubed integer while (x^2 -x*y +y^2) can equal a cubed integer, given that x and y are also integers. The conversation also includes an example to illustrate the difficulty in finding a case where both terms are cubed integers at the same time. The speaker is seeking help in understanding how to prove this.
  • #1
drd0013
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Okay, so I'm trying to figure something out.
Why can (x+y)=integer^3 not be true while
(x^2-x*y+y^2)=integer^3 is true?
where x and y are also integers. The integer to the right is just an undefined integer.
I'm trying to prove Fermat's last theorem x^n+y^n=z^n for n=3.
I was able to define two factors of x^2+y^2 as:
(x+y) and (x*y+(x-y)^2)) which reduces to (x^2-x*y+y^2).
Also, z^3 must be the product of cubed integers. So, (x+y) must itself be a cubed integer as well as (x^2-x*y+y^2) in order for x^3+y^3 to be = to z^3. (Which we know it can't). However, I don't know enough maths to prove that (x+y) cannot be equal to a cubed integer while (x^2-x*y+y^2) is.

Also, is this proof a novel one? I was just doing it for garbages and giggles.
Thanks to any who may be able to help
 
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  • #2
(x+y)*(x^2 -x*y +y^2) = z^3
the problem is not that x+y can't be a cubed value, it can, or that x^2 -x*y +y^2 can't be a cubed value, it can. it's proving that there is no case where both are cubed values for the same x,y.

let's take an example. 10^3 = 1000.
10^3 = 2^3 *5^3
so ether x+y = 8, or x+y = 125.
let;s assume the smaller one. which means x^2 -x*y +y^2 needs to equal 125.
x = 3 y= 5
3^2 - 3*5 +5^2 = 9 -15 +25 = 19; no where close to 125.
 
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  • #3
Yeah, I'm just not sure how (if it can be) to go about showing that. I'm trying to prove that the first and second terms can't both be equal to cubed integers at the same time. But I am not sure how to set up or solve a proof of that. I'd assume modus tolens of some kind but don't really know
 
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FAQ: Why two given equations cannot both be true (fermat's for n=3)

1. Why is it impossible for two equations to both be true in Fermat's Last Theorem for n=3?

The answer to this question lies in the fundamental theorem of arithmetic. This theorem states that every positive integer can be expressed as a unique product of prime numbers. In the case of n=3, the equation x³ + y³ = z³ would require three distinct prime factors for each variable. However, the equation xⁿ + yⁿ = zⁿ only requires two distinct prime factors for each variable. This means that there is no possible solution where both equations can be true simultaneously.

2. Can the equations be true for any values other than n=3?

No, Fermat's Last Theorem has been proven for all values of n greater than 2. This means that for any value of n greater than 2, there are no solutions where both equations xⁿ + yⁿ = zⁿ and xⁿ + yⁿ = zⁿ can be true at the same time.

3. Are there any exceptions to Fermat's Last Theorem for n=3?

No, there are no exceptions to Fermat's Last Theorem for n=3. This has been proven by Andrew Wiles in 1994 and is one of the most famous and well-known theorems in mathematics.

4. Is it possible for one of the equations to be true while the other is false?

No, both equations cannot be true simultaneously for any values of x, y, and z. This is because if one equation is true, it would mean that the other equation is also true, violating the fundamental theorem of arithmetic as mentioned in the answer to the first question.

5. What implications does Fermat's Last Theorem have on other mathematical problems?

Fermat's Last Theorem has had a significant impact on the study of number theory and has inspired the development of new techniques and theories. It also has important applications in cryptography and coding theory. Additionally, the proof of this theorem has also led to a better understanding of algebraic geometry and modular forms.

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