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Erdős–Moser baseline: odd exponents and congruence restrictions
Short result
The initial attempt excludes all odd-exponent solutions except (m,k)=(3,1) and derives congruence and size restrictions on any further solution. It does not resolve the even-exponent case or prove the conjecture.
Answer content
The response uses modular pairing to eliminate other odd-exponent solutions. For any further solution, it derives even k ≥ 2, m ≡ 3 (mod 8), restrictions modulo prime squares, and k < m log 2.
These are necessary conditions, not a proof of nonexistence. The remaining gap is to exclude every pair satisfying those conditions. The full argument is preserved in the attached transcript; no novelty claim is made.
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