A heuristic view on Fermat's Last Theorem using sums of consecutive odd numbers

Abstract: This paper presents a novel approach to explaining the validity of Fermat's Last Theorem and Euler's sum of powers conjecture through the internal architecture of numbers. The author utilizes a fundamental arithmetic principle: any natural number raised to the power k (Nᵏ) can be represented as a sum of N consecutive odd numbers. By analyzing these sequences, the paper demonstrates a critical distinction between the second power and all higher powers: For squares (k=2), the sequences are nested and continuous, starting from 1, which allows for the existence of Pythagorean triples (A² + B² = C²). For higher powers (k > 2), a "structural gap" emerges as the starting odd number of each sequence shifts forward at an accelerating rate, defined by the formula X = N^ (k-1) - (N-1). The author argues that the impossibility of decomposing a power into the sum of only two others for k > 2 is caused by a divergence in "nominal weight" (density) of the odd numbers. Even if the quantity of numbers in a sum is correct, their collective arithmetic mass from the beginning of the series cannot match the density required for a target block of a higher order. Furthermore, this model provides an arithmetic justification for Euler's conjecture, suggesting that at least k summands are required to "stitch" the structural gap inherent in the k-th degree. This theoretical framework serves as a foundation for the author's practical discovery of parametric series for "quadruples" of cubes (A³ = B³ + C³ + D³).