Erdős problem #366

Test the single-prime-support successor branch for structural information.

First action: Write the exact factor/gcd decomposition of p^e-1 for n+1=p^e, then enumerate e>=3 and p^e<=10^6 as a regression fixture against tools/erdos366_discriminator.py.

Stop or redirect when: Stop immediately on a verified witness; otherwise stop after the declared shell and retain the route only if it proves a local obstruction or material pruning rule.

• Analyze n+1=p^e using the factorization of p^e-1.
  Derive gcd conditions, implement a bounded certified shell, and compare with all single-prime-support cases through 10^6.
• Reconstruct the A060355 computation below 10^22.
  Audit OEIS history and Donovan Johnson table provenance for code and completeness arguments.
• Formalize a future explicit witness in Lean.
  After a witness exists, instantiate the existential and compile with no sorry or admit.

• isolated adversarial reconstruction
  Recover or independently rebuild the A060355 enumeration and verify both its coverage and every listed factorization without using the present sparse target code.
• symbolic and congruence family discovery
  Decompose p^e-1 into cyclotomic factors, exploit their controlled gcds, and test whether the powerfulness requirement forces forbidden exponent-one prime divisors.
• counterexample and witness search
  Generate every 3-full m in the bounded domain exactly once from increasing prime supports and exponents at least three, factor m-1 exactly, and independently compare the generated set with a dense sieve oracle.
  Reopen only if: Resume undifferentiated bound extension only with a recognized target bound, an explicit request, a completeness-certificate objective, or a new structural pruning result.

• Treat OEIS A060355 alone as a certified target exclusion through 10^22.
  It is a superset table and lacks an independently checked completeness certificate for the reported range.
  Reopen only if: Recover or reproduce the enumeration with a complete generator and independent certificate.
• Routine extension of the same arbitrary search cutoff as a contribution.
  A finite null search cannot settle universal nonexistence and would not improve a verified best-known result.
  Reopen only if: A predeclared recognized bound, explicit expert request, or new structural pruning theorem.
• Use (8,9), (12167,12168), or generic Pell-generated powerful pairs as target evidence without checking orientation and exponent three.
  They prove 3-full then 2-full, or only 2-full then 2-full.
  Reopen only if: Produce a family member whose first number independently passes 2-full and whose successor passes 3-full.