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Topic: Fermat revised ?
Replies: 8   Last Post: Aug 18, 2007 3:51 PM

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stephane.frion@gmail.com

Posts: 14
Registered: 8/15/07
Re: Fermat revised ?
Posted: Aug 17, 2007 10:45 PM
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> more clearly from x^2 + y^3 = z^4
> lets go to y^3 = z^4 - x^2 = (z^2 +x)(z^2 -x)
> now we can conclude, that number y should be
> built from two factors: let y = a*b
> and a^3 = z^2 + x
> b^3 = z^2 - x
> and then 2z^2 = a^3 + b^3................(*)
> 2x = a^3 - b^3
> and where only the eq(*) will determine some
> solution: we can notice for a=4 b=2;
> 2z^2 = 64 + 8 = 2*36 and z=6;x=28;y=8
> however even 28^2 + 8^3 = 6^4
> so it equals (2^4)(7^2) + 2^6 = (2^4)(3^4)
> and after extraction of 2^4 will be only:
> 7^2 + 2^2 = 3^4
> Also for to achieve primitive solutions it is to
> add as in Your developments: condition a and b as
> odd numbers...
> so far I used just to calculate set:
> x=433; y=13*11=143; z=42
> Thank You very much for more bigger samples.
> You mentioned also, that there is no certain method for
> to achieve such solutions ?
> Anyway to the author of this topic "stephane"
> should be said that there are no primitive solutions
> from n=3:
> any x^3 + y^4 equals z^5 for x;y;z of gcd=1 and etc. Also so called "Fermat revised"
> x^n + y^(n+1) = z^(n+2) conjecture for n=>3
> is just a part of well known Beal's conjecture...
> With the Best Regards
> Roman B. Binder
> POBox 3692 Jerusalem 91035 Israel
> RoBin
>


Thanks for your answer. Btw, I wasn't aware of the Beal's conjecture.
I will check it out certainly.


First, I wasn't really looking for solution which are relatively prime
( I just read in the web that if n=2 the equation has solution only if
x, y and z are relatively prime ... This is not the case as I could
see from the first answer...).In fact, I was simply looking for
solutions.
Since, I have able to find solutions for n=3
X Y Z
256 64 32
268912 19208 2744
209952 11664 1944
839808 23328 3888

I found out also that I could find solutions in negative
X Y Z
-128 -32 -16
-33614 -2401 -343
-69984 -3888 -648
-559872 -15552 -2592

I was unable to find pattern for these solutions.

SF
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