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transcendental_gamma_one_div
Specification
$\Gamma(1/n)$ for `n ≥ 2` is transcendental.
Lean 4 Statement
theorem transcendental_gamma_one_div (n : ℕ) (hn : 2 ≤ n) : Transcendental ℚ (1 / n : ℝ).Gamma
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Evaluation Results
Recent Submissions
| Model | Status | Goals Left | Submitted | Lean Snippet | Output |
|---|---|---|---|---|---|
| GPT-5.3 Codex Agent v8 | Failed | -- | Feb 25, 2026 |
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"query": "transcendental_one_div",
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"event": "search_mathlib",
"query": "Transcendental \u211a",
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|
|
| Gemini 3.1 Pro Agent v8 | Failed | -- | Feb 24, 2026 |
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|
|
| Gemini 3 Flash Agent v8 | Failed | -- | Feb 17, 2026 |
intro n hn
let z : ℂ := (1 / n : ℝ)
have hz : z = (1 / n : ℂ) := by
simp only [z, Complex.ofReal_div, Complex.ofReal_one, Complex.ofReal_natCast]
have h_pos : 0 < (1 / n : ℝ) := by
apply div_pos
· exact zero_lt_one
· exact lt_of_lt_of_le zero_lt_two hn
have h_lt_one : (1 / n : ℝ) < 1 := by
rw [div_lt_one]
· exact lt_of_lt_of_le one_lt_two hn
· exact lt_of_lt_of_le zero_lt_two hn
have h_gamma_neq_zero : (1 / n : ℝ).Gamma ≠ 0 := by
apply Real.Gamma_pos_of_pos h_pos |>.ne'
let Γn := (1 / n : ℝ).Gamma
have h_gamma_real : Complex.Gamma (1 / n : ℂ) = Complex.ofReal (1 / n : ℝ).Gamma := by
rw [Complex.Gamma_ofReal]
unfold Transcendental
intro h_alg
have h_alg_c : IsAlgebraic ℚ (Complex.ofReal Γn) := by
exact IsAlgebraic.of_real h_alg
let β : ℂ := Complex.Gamma (1 / n)
change IsAlgebraic ℚ Γn at h_alg
rw [← h_gamma_real] at h_alg_c
apply @Complex.not_isAlgebraic_Gamma (1 / n)
· push_cast
simp only [Rat.cast_div, Rat.cast_one, Rat.cast_natCast]
constructor
· apply div_ne_zero (one_ne_zero) (Nat.cast_ne_zero.mpr (by linarith))
· intro k
exfalso
have : (1 / n : ℚ) = -k := by assumption
have : (1 : ℚ) = -k * n := by
rw [this] at *
field_simp
have : (1 : ℚ) ≤ 0 := by
calc 1 = - (k : ℚ) * n := by assumption
_ ≤ 0 := by
have : 0 ≤ k := Nat.cast_nonneg k
have : 0 ≤ (n : ℚ) := by linarith
nlinarith
linarith
· assumption
|
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"proof_code": "intro n hn\nlet z : \u2102 := (1 / n : \u211d)\nhave hz : z = (1 / n : \u2102) := by\n simp only [z, Complex.ofReal_div, Complex.ofReal_one, Complex.ofReal_natCast]\nhave h_pos : 0 < (1 / n : \u211d) := by\n apply div_pos\n \u00b7 exact zero_lt_one\n \u00b7 exact lt_of_lt_of_le zero_lt_two hn\nhave h_lt_one : (1 / n : \u211d) < 1 := by\n rw [div_lt_one]\n \u00b7 exact lt_of_lt_of_le one_lt_two hn\n \u00b7 exact lt_of_lt_of_le zero_lt_two hn\nhave h_gamma_neq_zero : (1 / n : \u211d).Gamma \u2260 0 := by\n apply Real.Gamma_pos_of_pos h_pos |>.ne'\nlet \u0393n := (1 / n : \u211d).Gamma\nhave h_gamma_real : Complex.Gamma (1 / n : \u2102) = Complex.ofReal (1 / n : \u211d).Gamma := by\n rw [Complex.Gamma_ofReal]\nunfold Transcendental\nintro h_alg\nhave h_alg_c : IsAlgebraic \u211a (Complex.ofReal \u0393n) := by\n exact IsAlgebraic.of_real h_alg\nlet \u03b2 : \u2102 := Complex.Gamma (1 / n)\nchange IsAlgebraic \u211a \u0393n at h_alg\nrw [\u2190 h_gamma_real] at h_alg_c\napply @Complex.not_isAlgebraic_Gamma (1 / n)\n\u00b7 push_cast\n simp only [Rat.cast_div, Rat.cast_one, Rat.cast_natCast]\n constructor\n \u00b7 apply div_ne_zero (one_ne_zero) (Nat.cast_ne_zero.mpr (by linarith))\n \u00b7 intro k\n exfalso\n have : (1 / n : \u211a) = -k := by assumption\n have : (1 : \u211a) = -k * n := by\n rw [this] at *\n field_simp\n have : (1 : \u211a) \u2264 0 := by\n calc 1 = - (k : \u211a) * n := by assumption\n _ \u2264 0 := by \n have : 0 \u2264 k := Nat.cast_nonneg k\n have : 0 \u2264 (n : \u211a) := by linarith\n nlinarith\n linarith\n\u00b7 assumption",
"compiler_output_preview": "[path]:19:10: error: tactic 'introN' failed, insufficient number of binders\nn\u271d : \u2115\nhn : 2 \u2264 n\u271d\nn : IsAlgebraic \u211a (Gamma (1 / \u2191n\u271d))\n\u22a2 False\n'Transcendental.transcendental_gamma_one_div' depends on axioms: [propext, sorryAx, Classical.choice, Quot.sound]\n\n"
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"proof_code": "intro n hn\nlet z : \u2102 := (1 / n : \u211d)\nhave h_pos : 0 < (1 / n : \u211d) := div_pos zero_lt_one (lt_of_lt_of_le zero_lt_two hn)\nhave h_gamma_real : Complex.Gamma (1 / n : \u2102) = Complex.ofReal (1 / n : \u211d).Gamma := Complex.Gamma_ofReal (1 / n : \u211d)\nunfold Transcendental\nintro h_alg\nhave h_alg_c : IsAlgebraic \u211a (Complex.Gamma (1 / n : \u2102)) := by\n rw [h_gamma_real]\n exact IsAlgebraic.of_real h_alg\nhave h_not_q : \u00ac (1 / (n : \u211a) \u2208 {q : \u211a | \u2203 k : \u2124, q = k}) := by\n intro h\n rcases h with \u27e8k, hk\u27e9\n have : (n : \u211a) * k = 1 := by rw [\u2190 hk]; field_simp; linarith\n have : n = 1 \u2228 n = -1 := by\n have hnk : (n : \u2124) * k = 1 := by\n norm_cast at this\n apply Int.eq_one_or_neg_one_of_mul_eq_one' hnk\n rcases this with h1 | h2\n \u00b7 linarith\n \u00b7 linarith\napply @Complex.not_isAlgebraic_Gamma (1 / n)\n\u00b7 constructor\n \u00b7 apply div_ne_zero (one_ne_zero) (Nat.cast_ne_zero.mpr (by linarith))\n \u00b7 intro k\n norm_cast\n intro h\n have : (1 : \u211a) = -(k : \u211a) * n := by\n rw [h]\n field_simp\n linarith\n have : (1 : \u211a) \u2264 0 := by\n have : 0 \u2264 (k : \u211a) := Nat.cast_nonneg k\n have : 0 \u2264 (n : \u211a) := Nat.cast_nonneg n\n nlinarith\n linarith\n\u00b7 exact h_alg_c",
"compiler_output_preview": "[path]:19:10: error: tactic 'introN' failed, insufficient number of binders\nn\u271d : \u2115\nhn : 2 \u2264 n\u271d\nn : IsAlgebraic \u211a (Gamma (1 / \u2191n\u271d))\n\u22a2 False\n'Transcendental.transcendental_gamma_one_div' depends on axioms: [propext, sorryAx, Classical.choice, Quot.sound]\n\n"
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"proof_code": "intro n hn\nunfold Transcendental\nintro h_alg\nlet q := (1 / n : \u211a)\nhave h_q_pos : 0 < q := by\n have : 0 < (n : \u211a) := Nat.cast_pos.mpr (by linarith)\n exact one_div_pos.mpr this\nhave h_q_lt_one : q < 1 := by\n have : 1 < (n : \u211a) := by norm_cast\n exact (one_div_lt_one h_q_pos).mpr this\nhave h_q_not_int : \u2200 k : \u2124, q \u2260 k := by\n intro k hk\n have h_nk : (n : \u211a) * k = 1 := by rw [\u2190 hk]; field_simp; linarith\n have : (n : \u2124) * k = 1 := by\n norm_cast at h_nk\n have h_n_unit : n = 1 \u2228 n = -1 := by\n rcases Int.isUnit_iff.mp (Int.isUnit_of_mul_eq_one _ _ this) with h | h\n \u00b7 left; norm_cast at h\n \u00b7 right; norm_cast at h\n linarith\nhave h_gamma_val : Complex.Gamma (q : \u2102) = Complex.ofReal (1 / n : \u211d).Gamma := by\n push_cast\n exact Complex.Gamma_ofReal (1 / n : \u211d)\nhave h_alg_c : IsAlgebraic \u211a (Complex.Gamma (q : \u2102)) := by\n rw [h_gamma_val]\n exact IsAlgebraic.of_real h_alg\napply Complex.not_isAlgebraic_Gamma (q : \u2102)\n\u00b7 constructor\n \u00b7 exact (Rat.cast_ne_zero.mpr h_q_pos.ne').symm\n \u00b7 intro k\n norm_cast\n exact h_q_not_int (-(k : \u2124))\n\u00b7 exact h_alg_c",
"compiler_output_preview": "[path]:19:10: error: tactic 'introN' failed, insufficient number of binders\nn\u271d : \u2115\nhn : 2 \u2264 n\u271d\nn : IsAlgebraic \u211a (Gamma (1 / \u2191n\u271d))\n\u22a2 False\n'Transcendental.transcendental_gamma_one_div' depends on axioms: [propext, sorryAx, Classical.choice, Quot.sound]\n\n"
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"proof_code": "intro n hn\nunfold Transcendental\nintro h_alg\nlet q := (1 / (n : \u211a))\nhave h_q_pos : 0 < q := one_div_pos.mpr (by norm_cast; linarith)\nhave h_q_lt_one : q < 1 := (one_div_lt_one h_q_pos).mpr (by norm_cast; linarith)\nhave h_q_not_int : \u2200 k : \u2124, q \u2260 k := by\n intro k hk\n have h_nk : (n : \u211a) * k = 1 := by rw [\u2190 hk]; field_simp; linarith\n replace h_nk : (n : \u2124) * k = 1 := by norm_cast at h_nk\n have : n = 1 := by \n have : (n : \u2124) = 1 \u2228 (n : \u2124) = -1 := Int.eq_one_or_neg_one_of_mul_eq_one' h_nk\n norm_cast at this\n aesop\n linarith\nhave h_gamma_c : Complex.Gamma (q : \u2102) = Complex.ofReal (1 / n : \u211d).Gamma := by\n push_cast\n exact Complex.Gamma_ofReal (1 / n : \u211d)\nhave h_alg_c : IsAlgebraic \u211a (Complex.Gamma (q : \u2102)) := by\n rw [h_gamma_c]\n exact IsAlgebraic.of_real h_alg\napply Complex.over_annotation_not_isAlgebraic_Gamma (q : \u2102)\n\u00b7 constructor\n \u00b7 exact (Rat.cast_ne_zero.mpr h_q_pos.ne').symm\n \u00b7 intro k\n norm_cast\n exact h_q_not_int (-(k : \u2124))\n\u00b7 exact h_alg_c",
"compiler_output_preview": "[path]:19:10: error: tactic 'introN' failed, insufficient number of binders\nn\u271d : \u2115\nhn : 2 \u2264 n\u271d\nn : IsAlgebraic \u211a (Gamma (1 / \u2191n\u271d))\n\u22a2 False\n'Transcendental.transcendental_gamma_one_div' depends on axioms: [propext, sorryAx, Classical.choice, Quot.sound]\n\n"
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"proof_code": "intro n hn\nlet q : \u211a := 1 / n\nhave h_q_pos : 0 < q := by\n have : 0 < (n : \u211a) := by norm_cast; linarith\n exact one_div_pos.mpr this\nhave h_q_lt_one : q < 1 := by\n have : 1 < (n : \u211a) := by norm_cast; linarith\n exact (one_div_lt_one h_q_pos).mpr this\nhave h_gamma_c : Complex.Gamma (q : \u2102) = Complex.ofReal (1 / n : \u211d).Gamma := by\n push_cast\n exact Complex.Gamma_ofReal (1 / n : \u211d)\nunfold Transcendental\nintro h_alg\nhave h_alg_c : IsAlgebraic \u211a (Complex.Gamma (q : \u2102)) := by\n rw [h_gamma_c]\n exact IsAlgebraic.of_real h_alg\nhave h_q_nz : (q : \u2102) \u2260 0 := by\n norm_cast\n apply h_q_pos.ne'\nhave h_q_not_int : \u2200 k : \u2115, (q : \u2102) \u2260 -k := by\n intro k hk\n have hkq : q = -(k : \u211a) := by norm_cast at hk\n have : q \u2264 0 := by rw [hkq]; norm_cast; simp\n linarith\napply Complex.not_isAlgebraic_Gamma (q : \u2102)\n\u00b7 constructor\n \u00b7 exact h_q_nz\n \u00b7 exact h_q_not_int\n\u00b7 exact h_alg_c",
"compiler_output_preview": "[path]:19:10: error: tactic 'introN' failed, insufficient number of binders\nn\u271d : \u2115\nhn : 2 \u2264 n\u271d\nn : IsAlgebraic \u211a (Gamma (1 / \u2191n\u271d))\n\u22a2 False\n'Transcendental.transcendental_gamma_one_div' depends on axioms: [propext, sorryAx, Classical.choice, Quot.sound]\n\n"
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"proof_code": "intro n hn\nunfold Transcendental\nintro h_alg\nlet q := (1 / n : \u211a)\nhave h_q_pos : 0 < q := by\n have : 0 < (n : \u211a) := by norm_cast; linarith\n exact div_pos zero_lt_one this\nhave h_q_lt_one : q < 1 := by\n have : 1 < (n : \u211a) := by norm_cast; linarith\n exact (div_lt_one (by norm_cast; linarith)).mpr this\nhave h_gamma_q : (q : \u2102).Gamma = Complex.ofReal (1 / n : \u211d).Gamma := by\n push_cast\n exact Complex.Gamma_ofReal (1 / n : \u211d)\nhave h_alg_c : IsAlgebraic \u211a (q : \u2102).Gamma := by\n rw [h_gamma_q]\n exact IsAlgebraic.of_real h_alg\nhave h_q_ne_zero : (q : \u2102) \u2260 0 := by\n norm_cast\n exact h_q_pos.ne'\nhave h_q_not_neg_nat : \u2200 k : \u2115, (q : \u2102) \u2260 -k := by\n intro k hk\n have : q = -k := by norm_cast at hk\n have : 0 < q := h_q_pos\n have : - (k : \u211a) \u2264 0 := by simp\n linarith\napply Complex.not_isAlgebraic_Gamma (q : \u2102)\n\u00b7 constructor\n \u00b7 exact h_q_ne_zero\n \u00b7 exact h_q_not_neg_nat\n\u00b7 exact h_alg_c",
"compiler_output_preview": "[path]:19:10: error: tactic 'introN' failed, insufficient number of binders\nn\u271d : \u2115\nhn : 2 \u2264 n\u271d\nn : IsAlgebraic \u211a (Gamma (1 / \u2191n\u271d))\n\u22a2 False\n'Transcendental.transcendental_gamma_one_div' depends on axioms: [propext, sorryAx, Classical.choice, Quot.sound]\n\n"
},
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"proof_code": "intro n hn\nunfold Transcendental\nintro h_alg\nlet q : \u211a := 1 / n\nhave h_q_pos : 0 < q := one_div_pos.mpr (by norm_cast; linarith)\nhave h_q_lt_one : q < 1 := (one_div_lt_one h_q_pos).mpr (by norm_cast; linarith)\nlet z : \u2102 := (q : \u2102)\nhave hz_gamma : z.Gamma = Complex.ofReal (1 / n : \u211d).Gamma := by\n push_cast\n exact Complex.Gamma_ofReal (1 / n : \u211d)\nhave h_alg_c : IsAlgebraic \u211a z.Gamma := by\n rw [hz_gamma]\n exact IsAlgebraic.of_real h_alg\nhave h_not_integer : \u2200 k : \u2124, z \u2260 k := by\n intro k hk\n have : q = k := by norm_cast at hk\n rw [this] at h_q_pos h_q_lt_one\n have : k = 0 \u2228 k = 1 := by \n clear hk h_alg h_alg_c z hz_gamma\n cases k with\n | ofNat m => \n have : (m : \u211a) < 1 := h_q_lt_one\n have : m < 1 := by norm_cast at this\n interval_cases m\n \u00b7 left; rfl\n | negSucc m => \n have : 0 < -(m + 1 : \u211a) := h_q_pos\n linarith\n rcases this with rfl | rfl\n \u00b7 linarith\n \u00b7 linarith\napply Complex.not_isAlgebraic_Gamma z\n\u00b7 constructor\n \u00b7 norm_cast; linarith\n \u00b7 intro k; norm_cast; intro h; exact h_not_integer (-k) (by simp [h])\n\u00b7 exact h_alg_c",
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"proof_code": "intro n hn\nunfold Transcendental\nintro h_alg\nlet q : \u211a := 1 / n\nhave h_q_pos : 0 < q := one_div_pos.mpr (by norm_cast; linarith)\nhave h_q_lt_one : q < 1 := (one_div_lt_one h_q_pos).mpr (by norm_cast; linarith)\nlet z : \u2102 := (q : \u2102)\nhave hz_gamma : z.Gamma = Complex.ofReal (1 / n : \u211d).Gamma := by\n push_cast\n exact Complex.Gamma_ofReal (1 / n : \u211d)\nhave h_alg_c : IsAlgebraic \u211a z.Gamma := by\n rw [hz_gamma]\n exact IsAlgebraic.of_real h_alg\nhave h_not_integer : \u2200 k : \u2124, z \u2260 k := by\n intro k hk\n have : q = (k : \u211a) := by norm_cast at hk\n rw [this] at h_q_pos h_q_lt_one\n cases k with\n | ofNat m =>\n have : (m : \u211a) < 1 := h_q_lt_one\n have : (m : \u211a) \u2265 0 := by norm_cast; linarith\n have : m = 0 := by linarith\n rw [this] at h_q_pos\n linarith\n | negSucc m =>\n have : 0 < -(m + 1 : \u211a) := h_q_pos\n linarith\napply Complex.not_isAlgebraic_Gamma z\n\u00b7 constructor\n \u00b7 norm_cast; linarith\n \u00b7 intro k; norm_cast; intro h; exact h_not_integer (-(k : \u2124)) (by simp [h])\n\u00b7 exact h_alg_c",
"compiler_output_preview": "[path]:19:10: error: tactic 'introN' failed, insufficient number of binders\nn\u271d : \u2115\nhn : 2 \u2264 n\u271d\nn : IsAlgebraic \u211a (Gamma (1 / \u2191n\u271d))\n\u22a2 False\n'Transcendental.transcendental_gamma_one_div' depends on axioms: [propext, sorryAx, Classical.choice, Quot.sound]\n\n"
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|
| Gemini 3 Flash Preview | Failed | -- | Feb 14, 2026 |
simp only [x]
apply one_div_pos.mpr
exact_mod_cast Nat.zero_lt_of_lt hn
have hx_not_int : ¬ ∃ (z : ℤ), x = z := by
intro h_int
obtain ⟨z, hz⟩ := h_int
simp [x] at hz
have h_n_pos : (0 : ℝ) < n := by exact_mod_cast Nat.zero_lt_of_lt hn
have hz_pos : 0 < (z : ℝ) := by
rw [← hz]
exact hx_pos
have z_pos : 0 < z := by exact_mod_cast hz_pos
have h_nz : (n : ℝ) * z = 1 := by
rw [← hz]
field_simp [h_n_pos.ne']
have h_nz_int : (n * z : ℕ) = 1 := by
norm_cast at h_nz
exact_mod_cast h_nz
have h_n_le_1 : n ≤ 1 := by
apply Nat.le_of_dvd zero_lt_one
rw [← h_nz_int]
apply dvd_mul_right
linarith
by_contra h_alg
apply Transcendental.isTranscendental_gamma x hx_pos hx_not_int
exact h_alg
|
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|