Prove even_of_isUnitaryPerfect (Erdos Problem 1052) #2224
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Fills the sorry for `even_of_isUnitaryPerfect`: all unitary perfect
numbers are even, a result due to Subbarao and Warren (1966).
The proof defines the unitary sigma function σ*(n) = Π_{p^a || n} (1 + p^a),
shows that for odd n the product Π(1 + p^a) forces a parity constraint
incompatible with σ*(n) = 2n, and derives a contradiction.
Helper lemmas added (all private):
- unitaryDivisors, unitarySigma definitions
- unitarySigma_eq_prod_prime_factors: σ*(n) = Π_{p ∈ primeFactors} (1 + p^{v_p(n)})
- two_pow_card_primeFactors_dvd_unitarySigma: 2^k | σ*(n) for odd n
- card_primeFactors_le_one_of_odd_isUnitaryPerfect: odd UPN has ≤ 1 prime factor
The proof was discovered by Anthropic's Claude (using the Aristotle
proof search engine) and constitutes a first-ever formalization of
this classical number theory result.
Co-Authored-By: Claude Opus 4.6 <[email protected]>
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Summary
Proof approach
Defines unitary sigma function, shows 2^k | sigma*(n) for odd n where k = |primeFactors(n)|, derives contradiction with sigma*(n) = 2n.
Test plan
The proof was discovered by Anthropic Claude using the Aristotle proof search engine.