In simple mathematical notation:
Xn + Yn ≠ Zn if X, Y, Z, and n are integers and n ≥3.
For ease of expression in this discussion, I shall call the equation Xn + Yn = Zn Fermat’s equation. Note that the ‘equals’ sign (=) is replaced by ≠ (is not equal to) if Fermat’s Conjecture is true. Fermat’s proofs for the case n = 3 emerged from his writings after his death, but no general proof for all n > 2 was ever found.
FLT is a statement about numbers that is easy to state and easy to understand, but it went without proof for more than 300 years, even though the best and brightest mathematicians tried to prove (or disprove) it. Professional mathematicians correctly labeled Fermat’s Last Theorem a conjecture, which is what a mathematical statement should be called until it is either proved or disproved. It is likely that every mathematician alive in the last three centuries has tried to prove FLT, because the longer it went without resolution, the more famous it became, and resolving such a puzzle would assure the one who was first to solve it great recognition and fame; and mathematicians rank right up there with, or very close to, physicists, as a group of people with well-developed egos.
Karl Friedrich Gauss, arguably one of the greatest mathematicians who ever lived, was no exception. He invented modular algebra as a tool for use to investigate Diophantine equations, which is what Fermat’s equation is. [Diophantus specialized in solving equations with whole-number (integer) solutions. So, mathematicians call equations for which only integer solutions are sought, Diophantine equations.] Gauss proved Fermat’s Conjecture for n = 3, using modular algebra and complex numbers, but failed to be able to generalize the proof to greater values of n. When asked about his interest in Fermat’s Conjecture by Astronomer Wilhelm Olbers, Gauss said:
“I confess that Fermat's Theorem as an isolated proposition has very little interest for me, because I could easily lay down a multitude of such propositions, which one could neither prove nor dispose of.”
Could this disdain for FLT be the statement of a big ego, unwilling to admit that he couldn’t prove it, even though he tried? Probably so. Even a great man like Gauss can be blinded by his own brilliance. But I think Gauss can be forgiven his short-sightedness, because many brilliant thinkers believe that there are mathematical statements that can never be proved or disproved, and that FLT is an abstract theorem with little or no practical application outside of abstract number theory. Modern mathematicians, on the other hand, who believe the same things, should not be let off the hook so easily, because they have Gӧdel’s Incompleteness Theorems, which sheds some light on the questions of proof and provability.
The fact is however, that Gӧdel’s Incompleteness Theorems do not say that there are propositions that can never be proved. They only say that there are propositions that cannot be proved using only the system of logic within which they were stated. Every logical system is based on one or more a priori assumptions. A meaningful statement that cannot be proved true or false within a given logical system, may actually be provable when the a priori assumptions of the system are either added to, or changed in some fundamental way.
Conclusion: All of the legitimate questions raised by reviewers of FLT over the years have been eliminated and resolved. Therefore: