Analysis of the nature of the roots of functional equations and proof of their irrationality, which resulted in the proof of Fermat's theorem

This manuscript presents a novel analytical approach to Fermat’s Last Theorem (FLT). We introduce a functional analysis framework defining a function f(t) = aᵗ + bᵗ - 1, where a, b ∈ Q. By establishing the continuity and strict monotonicity of $f(t)$, we prove the existence of a unique root t₀ in the interval (0, +∞). We demonstrate that if t₀ were to coincide with any natural number $n > 2$, it would lead to an arithmetic contradiction based on the Fundamental Theorem of Arithmetic. Specifically, using p-adic valuation (vₚ) and the Lifting The Exponent (LTE) Lemma, we show that the p-adic density of the left-hand side (xᵗ + yᵗ) is strictly lower than that of the right-hand side (zᵗ), creating a "structural deficit." Furthermore, we employ binomial expansion and modular arithmetic to confirm that this imbalance persists regardless of the parity of the exponent. We conclude that Fermat’s equation has no solutions in natural numbers.