The assertion that for any integer exponent n>2 there are no positive integers a,b,c satisfying a^n + b^n = c^n; the unique nontrivial integer solutions occur only for n=1 and n=2.

Principle

A global impossibility statement about Diophantine equations of pure power degree greater than two: no nonzero integer triple can realize a nontrivial sum of like powers when the exponent exceeds two.

Demonstration

Concrete domain example: for n=3 one searches for positive integer solutions to a^3 + b^3 = c^3 and finds none; historically this specific case and others were studied until a general proof established that no such triple exists for any integer n>2 by connecting hypothetical solutions to impossible properties of certain elliptic curves and modular forms.

Misapplication

Claiming the theorem forbids real or rational solutions, or using it to exclude solutions for n=2 (Pythagorean triples). Also misusing it to assert that every Diophantine equation of higher degree has no integer solutions ignores many solvable higher-degree equations.

Consequence

When applied correctly, it implies that any search for positive-integer solutions to a^n + b^n = c^n for n>2 is futile; historically it motivated deep developments in algebraic number theory, modularity, and the study of elliptic curves and Galois representations.

Reversal

The inversion is the case n=2, where infinitely many positive-integer solutions (Pythagorean triples) exist; changing the domain to rational numbers also admits infinitely many nontrivial solutions for n>2 (for example parametric families when allowing zero or sign changes).

Boundary

Scope: integers a,b,c>0 and integer exponent n>2. Excludes zero, sign variants, and solutions in other rings or fields; extensions and analogues (e.g., over finite fields or for other exponent types) lie outside the theorem's claim.

Semantic Tension

Tension arises between the statement 'no integer solutions' and related but distinct notions like 'no primitive solutions' or results about modular forms; another nearby meaning is the Diophantine study of higher-degree equations which may or may not admit solutions depending on structure beyond mere exponent size.

Synthesis

Fermat's Last Theorem is a precise negative claim about positive-integer solutions to homogeneous sum-of-like-powers equations for exponents above two; its proof required reframing the Diophantine problem into statements about elliptic curves and modularity, thereby uniting a simple elementary-sounding statement with deep modern algebraic machinery.