by_cases h : Even n
· exact h
· exfalso
have h_odd : Odd n := by
cases Nat.even_or_odd n with
| inl h_even => contradiction
| inr h_odd => exact h_odd
-- Now we have h_odd : Odd n and hn : Perfect n
-- Since the Odd Perfect Number Conjecture is an open problem,
-- we leave the goal False.
View
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"proof_code": "by_cases h_even : Even n\n\u00b7 exact h_even\n\u00b7 have h_odd : Odd n := Nat.odd_iff_not_even.mpr h_even\n rcases h_odd with \u27e8k, rfl\u27e9\n unfold Perfect at hn\n simp [Even]\n intro m\n have h_eq : 2 * k + 1 = 2 * m := by assumption\n have h_mod : (2 * k + 1) % 2 = (2 * m) % 2 := by rw [h_eq]\n norm_num at h_mod",
"compiler_output_preview": "[path]:31:26: error: unknown constant 'Nat.odd_iff_not_even.mpr'\n[path]:33:11: error: ambiguous term, use fully qualified name, possible interpretations [@_root_.Perfect, Nat.Perfect]\n'OddPerfectNumber.odd_perfect_number_conjecture' depends on ax\n...<truncated>"
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"proof_code": "cases Nat.even_or_odd n with\n| inl h_even => exact h_even\n| inr h_odd =>\n exfalso\n have h_perfect : _root_.Perfect n := hn\n -- The Odd Perfect Number Conjecture\n -- Since we cannot prove it, we just provide the setup.\n unfold _root_.Perfect at h_perfect\n -- Here we have h_odd : Odd n and h_perfect : sigma n = 2 * n (possibly)\n -- The remaining goal is False.",
"compiler_output_preview": "[path]:33:41: error: type mismatch\n hn\nhas type\n n.Perfect : Prop\nbut is expected to have type\n _root_.Perfect ?m.2250 : Prop\n[path]:33:36: error: Application type mismatch: In the application\n _root_.Perfect n\nthe argument\n n\nhas type\n \u2115 :\n...<truncated>"
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"proof_code": "by_cases h : Even n\n\u00b7 exact h\n\u00b7 unfold Perfect at hn\n -- At this point, we have hn: Perfect n and h: \u00acEven n.\n -- The goal is to prove Even n, which is a contradiction.\n -- This is the odd perfect number conjecture.\n exfalso\n apply h\n -- Now we need to prove Even n from hn.\n -- This is the hard part.",
"compiler_output_preview": "[path]:31:11: error: ambiguous term, use fully qualified name, possible interpretations [@_root_.Perfect, Nat.Perfect]\n'OddPerfectNumber.odd_perfect_number_conjecture' depends on axioms: [propext, sorryAx, Classical.choice, Quot.sound]\n\n"
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"proof_code": "by_cases h : Even n\n\u00b7 exact h\n\u00b7 exfalso\n have h_odd : Odd n := by\n cases Nat.even_or_odd n with\n | inl h_even => contradiction\n | inr h_odd => exact h_odd\n -- Now we have h_odd : Odd n and hn : Perfect n\n -- Since the Odd Perfect Number Conjecture is an open problem,\n -- we leave the goal False.",
"compiler_output_preview": "[path]:31:2: error: unsolved goals\ncase neg\nn : \u2115\nhn : n.Perfect\nh : \u00acEven n\nh_odd : Odd n\n\u22a2 False\n'OddPerfectNumber.odd_perfect_number_conjecture' depends on axioms: [propext, sorryAx, Classical.choice, Quot.sound]\n\n"
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"proof_code": "by_cases h : Even n\n\u00b7 exact h\n\u00b7 exfalso\n have h_odd : Odd n := by\n cases Nat.even_or_odd n with\n | inl h_even => contradiction\n | inr h_odd => exact h_odd\n have h_sigma : sigma n = 2 * n := by\n rw [Nat.Perfect] at hn\n try exact hn.2\n exact hn\n have h_mod : (2 * n) % 4 = 2 := by\n rcases h_odd with \u27e8k, rfl\u27e9\n rw [mul_add]\n norm_num\n rw [\u2190 h_sigma] at h_mod\n -- sigma n % 4 = 2",
"compiler_output_preview": "[path]:36:19: error: unknown identifier 'sigma'\n[path]:40:36: error: unsolved goals\ncase intro\nk : \u2115\nhn : (2 * k + 1).Perfect\nh : \u00acEven (2 * k + 1)\nh_sigma : sorry = 2 * (2 * k + 1)\n\u22a2 (2 * (2 * k) + 2) % 4 = 2\n[path]
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"proof_code": "by_cases h : Even n\n\u00b7 exact h\n\u00b7 exfalso\n have h_odd : Odd n := by\n cases Nat.even_or_odd n with\n | inl h_even => contradiction\n | inr h_odd => exact h_odd\n try unfold Nat.Perfect at hn",
"compiler_output_preview": "[path]:31:2: error: unsolved goals\ncase neg\nn : \u2115\nhn : \u2211 i \u2208 n.properDivisors, i = n \u2227 0 < n\nh : \u00acEven n\nh_odd : Odd n\n\u22a2 False\n'OddPerfectNumber.odd_perfect_number_conjecture' depends on axioms: [propext, sorryAx, Classical.choice, Quot.sound]\n\n"
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"proof_code": "by_cases h : Even n\n\u00b7 exact h\n\u00b7 exfalso\n have h_odd : Odd n := by\n cases Nat.even_or_odd n with\n | inl h_even => contradiction\n | inr h_odd => exact h_odd\n unfold Nat.Perfect at hn\n -- n is an odd perfect number.\n -- The sum of its proper divisors is n.\n -- This remains an open problem in number theory.",
"compiler_output_preview": "[path]:31:2: error: unsolved goals\ncase neg\nn : \u2115\nhn : \u2211 i \u2208 n.properDivisors, i = n \u2227 0 < n\nh : \u00acEven n\nh_odd : Odd n\n\u22a2 False\n'OddPerfectNumber.odd_perfect_number_conjecture' depends on axioms: [propext, sorryAx, Classical.choice, Quot.sound]\n\n"
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"proof_code": "by_cases h : Even n\n\u00b7 exact h\n\u00b7 exfalso\n have h_odd : Odd n := by\n cases Nat.even_or_odd n with\n | inl h_even => contradiction\n | inr h_odd => exact h_odd\n unfold Nat.Perfect at hn\n rcases hn with \u27e8h_sum, h_pos\u27e9\n -- Now we have n odd, n > 0, and the sum of its proper divisors is n.",
"compiler_output_preview": "[path]:31:2: error: unsolved goals\ncase neg.intro\nn : \u2115\nh : \u00acEven n\nh_odd : Odd n\nh_sum : \u2211 i \u2208 n.properDivisors, i = n\nh_pos : 0 < n\n\u22a2 False\n'OddPerfectNumber.odd_perfect_number_conjecture' depends on axioms: [propext, sorryAx, Classical.choice, Quot.sound]\n\n"
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"category": "research_open",
"endings": [
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"stdout": "[path]:31:2: error: unsolved goals\ncase neg\nn : \u2115\nhn : n.Perfect\nh : \u00acEven n\nh_odd : Odd n\n\u22a2 False\n'OddPerfectNumber.odd_perfect_number_conjecture' depends on axioms: [propext, sorryAx, Classical.choice, Quot.sound]\n",
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}
Gemini 3 Flash Preview
Partial
1
Feb 14, 2026
by_contra h_zero
subst h_zero
simp [Perfect] at hn
have h_odd_nat : n % 2 = 1 := Nat.odd_iff.mp (Nat.odd_iff_not_even.mpr h_odd)
have h_sigma : sigma 1 n = 2 * n := hn
let p := {p : ℕ | p.Prime ∧ p % 4 = 1 ∧ ∃ k, p^k ∣ n ∧ Odd k}
rcases Nat.eq_zero_or_pos n with rfl | hn_pos
· simp [Perfect] at hn
· have : Odd n := Nat.odd_iff_not_even.mpr h_odd
obtain ⟨p, k, m, hp, hk, hm, h_n_form⟩ := Euler_form_odd_perfect hn h_pos this
aesop
