Prove this simple fact?

Question

Prove that it is impossible to write
2^(n) = t^(n) - w^(n)
for n>1 having natural solutions for "t" & "w".

Answer 1

We are given

2ⁿ = tⁿ - wⁿ



Now, we transpose wⁿ to the left

2ⁿ + wⁿ = tⁿ



Now 2, w, t and n are all natural numbers, and n > 1.
Now, we can prove that it is impossible for n > 1 by proving 2 things:
a) It is impossible for n > 2.
b) It is impossible for n = 2.

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a) Now, if n > 2, then according to Fermat's Last Theorem, there are no integer solutions to that equation, QED.
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b) Now, if n = 2, then the equation becomes

2² + t² = w²



or

4 + t² = w²



Now, t² is positive, and t² + 4 needs to be a perfect square.
We can easily prove that t and w are either both odd or both even.
Thus, their difference is a multiple of 2.
Let it be 2x.
Now, 2x is positive.
Thus,
w - t = 2x

We transpose
w = t + 2x

We square both sides
w² = t² + 4tx + 4x²

We substitute w²
4 + t² = t² + 4tx + 4x²

Now, we cancel t²
4 = 4tx + 4x²

We divide by 4
tx + x² = 1

We solve for t
t = 1/x - x

Now, we know that both 1/x and x must be positive natural numbers.
The only possible value for x is 1.
Thus, x = 1, and 2x = 2.
Thus, the difference of t and w is 2.
But if their difference is 2, then
4 + t² = t² + 4t + 4

And
4 = 4t + 4

and
4t = 0

Thus,
t = 0

Which is not a natural number. Therefore, it is impossible for n = 2, QED.
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Therefore, it is impossible to write
2ⁿ = tⁿ - wⁿ
for n > 1 having natural solutions for t and w. QED

^_^

Answer 2

you would need to solve for a root of 2.

Answer 3

When its a fact why strain our brains prooving it.

Answer 4

2^(n) = t^(n) - w^(n)

write as
2^(n) + w^(n) = t^(n)

and we have smoe statement that is proved by wiles a coulpe of ears ago.

Answer 5

go to ur schhol and bore ut teacher

Answer 6

Sorry, this stumps me! wow

Answer 7

im good at math , but cant solve this~!

Answer 8

Cant get it