Cohoes–Erdős Bound: Diophantine Fractional Growth

This paper introduces and formalizes the Fraction Growth Conjecture (Cohoes–Erdős Bound), which establishes a sublinear bound on the number of unique denominators in optimal unit fraction decompositions. The conjecture proposes that as the denominator of a fraction increases, the number of terms required in its minimal decomposition grows at most logarithmically. The paper provides rigorous theoretical justification using asymptotic analysis, additive number theory, and refinements to classical divisor function estimates, particularly improving upon Tenenbaum’s results. Computational validation up to ten thousand confirms that the number of unique denominators remains sublinear, supporting the conjecture. Additionally, connections to Farey sequences, Dirichlet divisor bounds, and Diophantine approximation are explored, along with broader implications for cryptography, combinatorial optimization, and computational number theory. This work advances our understanding of unit fraction growth constraints and offers a novel refinement of classical asymptotic divisor estimates.