The Math Forum

Search All of the Math Forum:

Views expressed in these public forums are not endorsed by NCTM or The Math Forum.

Math Forum » Discussions » sci.math.* » sci.math

Topic: ---- --- --- --- validity of an equation
Replies: 5   Last Post: Mar 14, 2013 6:20 PM

Advanced Search

Back to Topic List Back to Topic List Jump to Tree View Jump to Tree View   Messages: [ Previous | Next ]

Posts: 12,012
Registered: 7/15/05
Re: ---- --- --- --- validity of an equation
Posted: Mar 2, 2013 9:24 AM
  Click to see the message monospaced in plain text Plain Text   Click to reply to this topic Reply

Deep wrote:

>Consider the following equation under the given conditions
>y^(1/2) = (x^10 - z^ 5)^(1/5 ) (1)
>Conditions: y, x, z are co prime integers each > 5, 2|y.
>Question: can (1) have any solution ?

There are no solutions.

Suppose otherwise.

Raising both sides to the 10'th power yields

y^5 = (x^10 - z^5)^2

Letting w = x^10 - z^5, we get

y^5 = w^2

Since w^2 is a perfect 5'th power, and since gcd(2,5) = 1,
it follows that w is a perfect 5'th power. Thus, we can
write w = t^5 for some integer t.

The hypothesis implies that w is nonzero, hence t is also
nonzero. Then

w = x^10 - z^5

=> t^5 = x^10 - z^5

contrary to Fermat's Last Theorem for exponent 5.


Point your RSS reader here for a feed of the latest messages in this topic.

[Privacy Policy] [Terms of Use]

© The Math Forum at NCTM 1994-2017. All Rights Reserved.