Jacob's ladders, almost exact decomposition of certain increments of the Hardy-Littlewood integral (1918) by means of the Raabe's integral and the thirteenth equivalent of the Fermat-Wiles theorem

TL;DR

This paper introduces a novel approach using Jacob's ladders and Raabe's integral to derive the thirteenth equivalent of Fermat-Wiles theorem and achieve near-exact decomposition of specific Hardy-Littlewood integral increments.

Contribution

It presents a new method connecting Jacob's ladders with Raabe's integral to advance understanding of Fermat-Wiles theorem equivalents and Hardy-Littlewood integral decompositions.

Findings

Derivation of the thirteenth equivalent of Fermat-Wiles theorem

Almost exact decomposition of Hardy-Littlewood integral increments

New application of Jacob's ladders in number theory

Abstract

In this paper we use our theory of Jacob's ladders on the Raabe's integral to obtain: (i) The thirteenth equivalent of the Fermat-Wiles theorem, as well as (ii) almost exact decomposition of certain elements of continuum set of increments of the Hardy-Littlewood integral.