# A Diophantine Equation with no solutions

#### e2m2a

Does anyone know if this Diophantine equation is impossible to solve for all values of s?

C^u = A^u + B^s, where u = 2s

For example, is there no integer solution for C,A, and B for the following:

C^22 = A^22 + B^11

#### BvU

Homework Helper
It is possible to solve $C^2 = A^2 + B$ with integers for s = 1

#### e2m2a

But what about for s > 1? Are there no solutions and if so, is there a proof for this?

#### BvU

Homework Helper
Are the restrictions going to come in one at a time or is there a complete problem description ?
'Integers' is quite a bit more than 'natural numbers > 2' $\qquad$  $\ge 2$

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#### e2m2a

Are the restrictions going to come in one at a time or is there a complete problem description ?
'Integers' is quite a bit more than 'natural numbers > 2'
Interested only in positive integers or natural numbers for s > 1,

#### mfb

Mentor
A solution to C22 = A22 + B11 would be a solution to D11 = E11 + B11 where D = C2 and E = A2, which is impossible per Fermat's last theorem. Similar for all other positive integers s apart from 1.

BvU

#### e2m2a

A solution to C22 = A22 + B11 would be a solution to D11 = E11 + B11 where D = C2 and E = A2, which is impossible per Fermat's last theorem. Similar for all other positive integers s apart from 1.
Ok. That makes sense. Thanks.

"Diophantine Equation with no solutions"

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