can you proof this?

Question

Proof that the following equation

x^p + y^q = z^r

has no integer solution x,y,z for any p,q,r >2 integers

No this is NOT Fermats last theorem.
In Fermats the exponents are all equal.
Her they are are NOT p!=q!=r

Answer 1

This is a special case of the Fermat-Catalan conjecture. It is Tijdeman and Zagier who have conjectured that there are no solutions where all p,q,r > 2, but their conjecture is unproven.

The web page below shows the NINE non-trivial solutions known where one of p,q,r is 2 and the others are larger.

Answer 2

Ah, yes, Fermat's Last Theorem. The proof is a little too long to fit here. I'll refer you to Wiles' book.

Fermat's Last Theorem is notorious enough to have made several appearances in pop culture. For example, an incorrect counterexample appears in an episode of The Simpsons, "Treehouse of Horror VI". In the three-dimensional world in "Homer3", the equation 1782^12 + 1841^12 = 1922^12 is visible. The joke is that the twelfth root of the sum does evaluate to 1922 due to rounding errors when entered into most handheld calculators. In fact, the left hand sum evaluates to 2,541,210,258,614,589,176, 288,669,958,142,428,526,657, while the right hand side evaluates to 2,541,210,259,314,801,410, 819,278,649,643,651,567,616 — within a billionth of each other but still out by 700 octillion.

Answer 3

Fermat's last theorem says that x^n + y^n = z^n

has no (non-zero) integer solution x,y,z for n>2

So your case has as a particular case Fermat's last theorem. In other words, if this had aldready been proved (d not f) then Andrew Wiles would probably habe noticed and his proof of Fermat's theorem would have been of no need.

Answer 4

Its already been "proofed"