You say in the introduction that there is just one solution “apart from a few degenerate solutions in which x or y is zero”.

That is not exactly true: in fact there are no solutions with x=0 or y=0! I think that what you meant is that the only other solution is x=1, y=-1, which is quite trivial and sees the number 0 sandwiched (x=-1=y is a solution to the second equation, but not to the original problem).

This solution did not appear in your exposition because it arises from the consideration of the cases in which y has no prime factors, i.e., when y=1 or y=-1.

Regards,

Jose Brox

I have a hunch that it is to do with the fact that complex numbers do not obey the unique factorization principle i.e the fact that there is a unique way of factorizing a number into prime factors. I guess the above proof dabbled with trying to factorize into complex factors and hence may be faced with proving the rigor of this process. Correct me if I am wrong!

]]>Also, some of the more computational aspects of abstract algebra are useful in coding theory.

]]>