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Elementary Math Fractions, Percentages, Word Problems, Equations, Inequations, Factorization, Expansion


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September 2nd, 2017, 07:19 PM   #1
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my problem

Hey!
(x+1)^(n+1)-(x-1)^(n+1)=y^n
There is no solutions in total numbers x, y, n.
n>1
How to show it?
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September 3rd, 2017, 05:22 AM   #2
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"total numbers"? Do you mean "integers"?
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September 3rd, 2017, 05:38 AM   #3
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Yes.
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October 17th, 2017, 07:47 AM   #4
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look up Fermat's last theorem proof.
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October 17th, 2017, 09:05 AM   #5
SDK
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Math Focus: Dynamical systems, analytic function theory, numerics
Try a simple example first.

If $n=2$ then you want to prove that
\[(x+1)^3 - (x-1)^3 = y^2 \] has no integral solutions.
Suppose it did, then expanding gives us $6x^2 + 2 =y^2$. Do you see why this can't be true? Hint: compute this modulo 3.

Now apply a similar idea to the general expression using the binomial theorem.
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