Will you help me with an extra credit problem?

Question

My teacher gave me an extra credit proof and if anyone can get it he would pass us.

The question is to prove that a^x+ b^x = c^x where a, b, and c are constants and this statement cannot be true for x > 3

THANKS

Answer 1

The answers above are correct, this is Fermat's Last Theorem. However it is stated incorrectly. a,b,and c must be nonzero integers, not just constants. If you allow a,b, and c to be real numbers, there are an infinite number of solutions for x>3. Just choose 3 numbers where p+q = r and let a=p^(1/x), b=q^(1/x) and c=r^(1/x)

Answer 2

Counter-proof!
Let a = 0, and b = c for any value of c, and let x be any value > 3.
b^x = b^x
b^x = c^x
0+b^x = c^x
0^x + b^x = c^x
a^x + b^x = c^x

However, your teacher probably meant a non-zero solution.

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In that case, you are being asked to prove Fermat's Last Theorem!
Check out these pages for more details:

http://en.wikipedia.org/wiki/Fermat's_last_theorem
http://mathforum.org/dr.math/faq/faq.fermat.html
http://mathforum.org/library/drmath/view/52464.html (this link has a proof)

I think this is your teacher's idea as a joke, as the proof uses elliptic curves and modular forms, two quite complex branches of Mathematics. Good luck!

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However, being a little bit clever, you can avoid dealing with this. When writing proofs, one is allowed to use already-proved theorems, Fermat's Last Theorem, being one of them.

Prove: For any non-zero integers: a,b,c, there does not exist x>3 such that a^x+b^x = c^x.

Proof:
1. Suppose there exists x such that: a^x+b^x = c^x.
2. x > 3 (given)
3. 3 > 2
4. x > 2 (transitivity of >)
5. There does not exist x such that a^x+b^x = c^x (Fermat's Last Theorem)
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We have a contradiction. Thus, our assumption was false, in that x exists.

Answer 3

Your teacher is pullling your leg. This is one of the most famous math problems in existence, and is called Fermat's Last theorem - see http://primes.utm.edu/glossary/page.php?sort=FermatsLastTheorem - as this article mentions, a solution was developed 350 years later, in 1995. You might get some credit if you even mention it as Fermat's last theorem. Print out the article and take it in. Also print out http://www.pbs.org/wgbh/nova/proof/wiles.html - in that article, it says that the proof developed by Wiles is 150 pages long, so I don't think you want to go searching for that.