no three positive integers a, b, and c satisfy the equation an + bn = cn for any integer value of n greater than 2.

This statement is known as Fermat's Last Theorem. It was first proposed by Pierre de Fermat in 1637, and it remained an unsolved problem in mathematics for more than 350 years. The theorem was finally proven by the British mathematician Andrew Wiles in 1994.

Here's a step-by-step explanation of the theorem:

1. Let's assume that there are three positive integers a, b, and c that satisfy the equation a^n + b^n = c^n for an integer value of n greater than 2.

2. Fermat's Last Theorem states that no such triple of numbers can exist.

3. To understand why, consider the case where n=3. If a^3 + b^3 = c^3, then a, b, and c must be the sides of a right triangle. But the Pythagorean theorem (a^2 + b^2 = c^2) tells us that no such triangle can exist.

4. The same logic applies for all higher values of n. If a^n + b^n = c^n, then a, b, and c would have to be the sides of a "hyper-triangle" in n-dimensional space. But no such shape can exist.

5. Therefore, Fermat's Last Theorem is true: no three positive integers a, b, and c can satisfy the equation a^n + b^n = c^n for any integer value of n greater than 2.