Hard diophantine equation.

[tva_vlad]

Hello! I have a hard diophantine equation which I tried to solve last months but with no succes.
Prove that the equation x^2-y^10+z^5=6 has no integer solutions(positive, negative). I tried a modular approach(modulo 11)and it didn't work.
I tried to write it in several ways but again no chance. Do you have an idea? I would be very happy to see a COMPLETE proof. Many people say it works modulo 11 but I don't think so(almost 90% sure). I'm new to this forum

 

[eddybob123]

x^2 must be 0,1,3,4,5, or 9 mod 11
y^10 must be 1 mod 11 (By Fermat's Little Theorem)
z^5 must be 1 or -1 mod 11

It is not hard to see that x^2-y^10+z^5 cannot be 6 mod 11.

 

[tva_vlad]

x^2 must be 0,1,3,4,5, or 9 mod 11
y^10 must be 1 mod 11 (By Fermat's Little Theorem)
z^5 must be 1 or -1 mod 11

It is not hard to see that x^2-y^10+z^5 cannot be 6 mod 11.

I will use "=" for congruent. If x=4,7 mod 11, y=0 mod 11, z=1,3,4,5,9 mod 11 then x^2-y^10+z^5=6 mod 11 which you can check.