Explicit Bounds for the First Run of Consecutive Composite Integers

This note establishes explicit unconditional bounds for the first occurrence of \(n\) consecutive composite integers: \(A(n)<\exp(n/3)\) for \(n\ge2000\), and \(A(n)>25n(\log n)^2\) for \(n\ge400000\). The proofs combine a Chinese remainder theorem construction with explicit prime estimates of Pierre Dusart. Complete threshold calculations, references, and a discussion of limitations are included. These uniform bounds improve the elementary primorial construction but are nonoptimal and do not determine the true asymptotic growth of \(A(n)\). ChatGPT’s assistance is acknowledged in the paper.