Teen Proves New Lower Bound on Gaps Between Carmichael Numbers — E8 Intelligence Research

FINDING: High school student Daniel Larsen proved a key theorem about Carmichael numbers (pseudoprimes that pass Fermat's test for all bases), establishing a new lower bound on gaps between consecutive Carmichael numbers. | MATH: Let \(C(x)\) be the number of Carmichael numbers ≤ \(x\). Larsen's result: For sufficiently large \(x\), there exists a Carmichael number in every interval \([x, x + x1/3]\). This improves on previous bounds (e.g., \(x1/2\) or logarithmic gaps). Key constants: exponent \(1/3\) emerges from analytic number theory techniques (zero-density estimates for Dirichlet L-functions). | CONNECTION: No direct geometric ratio (0.382, 0.618, etc.) appears. However, the exponent \(1/3\) relates to cubic roots and threefold symmetry, which appears in crystallographic point groups (e.g., trigonal systems) and root systems of type \(A_2\). The result also uses sieve methods and harmonic analysis on number fields, which have deep ties to lattice structures. | DEPTH: 8 — A Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com