This post is motivated by this question. OP first stated the question as:
" Does Fermat's Last Theorem or Euler's proof for the case n = 3 imply that $x*a^3+y*b^3 = c^3$ has no solutions in integers for $a,b,c$ positive integers and integers $x$ and $y$? Thank you.."
There were answers adressing this. After the answers happened, he changed the assumptions in the comments:
"I should add the condition that a,b,c,x,y cannot be 1."
After the proper changings on answers, he further changed the hypotheses:
"and the gcd(a,b) = 1."
How to proceed on this behaviour?