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pi_pow_pi_pow_pi_pow_pi_not_integer
Specification
$\pi^{\pi^{\pi^\pi}}$ is not an integer. This would follow from $\pi^{\pi^{\pi^\pi}}$ being transcendental, but this formulation is of interest in its own right, as it could in principle be proven by direct computation. *Reference:* [YouTube](https://www.youtube.com/watch?v=BdHFLfv-ThQ)
Lean 4 Statement
theorem pi_pow_pi_pow_pi_pow_pi_not_integer : ¬ ∃ (n : ℤ), π ^ π ^ π ^ π = n
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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 |
View{
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{
"iteration": 1,
"event": "search_mathlib",
"query": "pi_pow_pi",
"results_count": 8
},
{
"iteration": 2,
"event": "search_mathlib",
"query": "Gelfond",
"results_count": 0
},
{
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"event": "search_mathlib",
"query": "transcendental_pi",
"results_count": 0
},
{
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"event": "search_mathlib",
"query": "irrational_pi",
"results_count": 1
},
{
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"event": "search_mathlib",
"query": "Irrational.rpow",
"results_count": 8
},
{
"iteration": 6,
"event": "search_mathlib",
"query": "Irrational.ne_int",
"results_count": 1
},
{
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"event": "search_mathlib",
"query": "Tactic.NormNum.irrational",
"results_count": 8
},
{
"iteration": 8,
"event": "search_mathlib",
"query": "Irrational.of_pow",
"results_count": 1
},
{
"iteration": 9,
"event": "search_mathlib",
"query": "pi_pow_pi_pow_pi_pow_pi_not_integer",
"results_count": 8
},
{
"iteration": 10,
"event": "search_mathlib",
"query": "pi_pow_pi_irrational",
"results_count": 8
}
]
}
|
|
| Gemini 3.1 Pro Agent v8 | Failed | -- | Feb 24, 2026 |
View{
"agent_telemetry": {
"compilations": 0,
"iterations": 0,
"duration_s": 0.04,
"best_status": "pending",
"best_goals": null,
"max_iterations": 10,
"model": "google/gemini-3.1-pro-preview",
"error": "OpenRouter API error 402: {\"error\":{\"message\":\"This request requires more credits, or fewer max_tokens. You requested up to 65536 tokens, but can only afford 58661. To increase, visit https://openrouter.ai/settings/keys and create a key with a higher total limit\",\"code\":402,\"metadata\":{\"provider_name\":null}},\"user_id\":\"user_2zsm3L1zzBL2yxyrvmrXwgeTimc\"}",
"provider": "openrouter_primary",
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"usage": {
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"response_trace": []
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|
|
| Gemini 3 Flash Agent v8 | Partial | 1 | Feb 17, 2026 |
intro h
rcases h with ⟨n, hn⟩
let x := π ^ (π ^ (π ^ π))
have hx : x = (n : ℝ) := hn
have h_pos : 0 < x := by
apply rpow_pos_of_pos
exact pi_pos
have n_pos : 0 < n := by
have : (0 : ℝ) < (n : ℝ) := by rwa [← hn]
exact_mod_cast this
have h_not_int : x ∉ Set.range (Int.cast : ℤ → ℝ) := by
intro h_mem
rcases h_mem with ⟨m, hm⟩
refine ?_
apply h_not_int
use n
exact hn.symm
|
View{
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"iterations": 8,
"duration_s": 81.39,
"best_status": "partial",
"best_goals": 1,
"max_iterations": 8,
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"error": null,
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"finish_reason": "stop",
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"cost_usd": 0.01404815
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"response_trace_count": 16
},
"response_trace": [
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"proof_code": "intro h\nrcases h with \u27e8n, hn\u27e9\nhave h_pos : 0 < \u03c0 ^ \u03c0 ^ \u03c0 ^ \u03c0 := by\n apply rpow_pos_of_pos\n apply rpow_pos_of_pos\n apply rpow_pos_of_pos\n exact pi_pos\nhave n_pos : 0 < n := by\n have : 0 < (n : \u211d) := by\n rw [\u2190 hn]\n exact h_pos\n exact_mod_cast this\nlet x := \u03c0 ^ \u03c0 ^ \u03c0 ^ \u03c0\nhave h_floor : (\u230ax\u230b : \u211d) \u2264 x \u2227 x < \u230ax\u230b + 1 := \u27e8floor_le x, lt_floor_add_one x\u27e9\nhave h_eq : x = (n : \u211d) := hn\nhave h_int : x = (\u230ax\u230b : \u211d) := by\n rw [h_eq]\n simp only [floor_intCast]\n norm_cast\nhave h_approx : 10 ^ 18 < x := by\n have h_pi_gt_3 : 3 < \u03c0 := pi_gt_three\n have h1 : 27 < \u03c0 ^ \u03c0 := by\n have : 3^3 < \u03c0^\u03c0 := rpow_lt_rpow (by linarith) h_pi_gt_3 (by linarith)\n norm_num at this\n exact this\n have h2 : 27^27 < \u03c0^\u03c0^\u03c0 := by\n apply rpow_lt_rpow (by linarith) (by linarith) h_pi_gt_3\n have h3 : (27^27)^3 < x := by\n apply rpow_lt_rpow (by linarith) (by linarith) h_pi_gt_3\n linarith\nhave h_not_int : x - \u230ax\u230b \u2260 0 := by\n unfold x\n intro h_zero\n let val := \u03c0 ^ \u03c0 ^ \u03c0 ^ \u03c0\n refine ?_\napply h_not_int\nrw [h_int]\nsimp",
"compiler_output_preview": "[path]:23:4: error: tactic 'apply' failed, could not unify the conclusion of `@rpow_pos_of_pos`\n 0 < ?x ^ ?y\nwith the goal\n 0 < \u03c0\n\nNote: The full type of `@rpow_pos_of_pos` is\n \u2200 {x : \u211d}, 0 < x \u2192 \u2200 (y : \u211d), 0 < x ^ y\ncase hx\nn : \u2124\nhn : \u03c0 ^ \u03c0 ^ \u03c0 ^ \u03c0 = \u2191n\n\u22a2 0 < \u03c0\n[path]
},
{
"iteration": 2,
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"finish_reason": "stop",
"response_id": "gen-1771350467-gL637CO9lYli57ACZ05U",
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"usage": {
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"proof_code": "intro h\nrcases h with \u27e8n, hn\u27e9\nhave h_pos : 0 < \u03c0 ^ \u03c0 ^ \u03c0 ^ \u03c0 := by\n apply rpow_pos_of_pos\n apply rpow_pos_of_pos\n apply rpow_pos_of_pos\n exact pi_pos\nhave n_pos : 0 < n := by\n have : 0 < (n : \u211d) := by rw [\u9ad8\u5ea6_eq_hn:]; exact h_pos\n exact_mod_cast this\n where \u9ad8\u5ea6_eq_hn : (n : \u211d) = \u03c0 ^ \u03c0 ^ \u03c0 ^ \u03c0 := hn.symm\nlet x := \u03c0 ^ \u03c0 ^ \u03c0 ^ \u03c0\nhave h_int : x = (n : \u211d) := hn\nhave : x \u2208 Set.range (Int.cast : \u2124 \u2192 \u211d) := by\n use n\n exact hn.symm\n-- The transcendental nature is not yet proven in Mathlib, leaving the remaining goal.\n-- We use the contradiction setup to indicate progress.\nsuffices : x \u2209 Set.range (Int.cast : \u2124 \u2192 \u211d)\n\u00b7 contradiction\nrefine ?_",
"compiler_output_preview": "[path]:27:33: error: expected token\n[path]:23:4: error: tactic 'apply' failed, could not unify the conclusion of `@rpow_pos_of_pos`\n 0 < ?x ^ ?y\nwith the goal\n 0 < \u03c0\n\nNote: The full type of `@rpow_pos_of_pos` is\n \u2200 {x : \u211d}, 0 <\n...<truncated>"
},
{
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"proof_code": "intro h\nrcases h with \u27e8n, hn\u27e9\nlet x := \u03c0 ^ \u03c0 ^ \u03c0 ^ \u03c0\nhave hx : x = (n : \u211d) := hn\nhave x_pos : 0 < x := by\n apply rpow_pos_of_pos\n apply rpow_pos_of_pos\n apply rpow_pos_of_pos\n exact pi_pos\nhave n_pos : 0 < n := by\n have : (0 : \u211d) < (n : \u211d) := by rwa [\u2190 hn]\n exact_mod_cast this\nhave h_not_int : x \u2209 Set.range (Int.cast : \u2124 \u2192 \u211d) := by\n intro range_h\n rcases range_h with \u27e8m, hm\u27e9\n have : (n : \u211d) = (m : \u211d) := by rwa [\u2190 hn]\n have : n = m := by exact_mod_cast this\n refine ?_\napply h_not_int\nuse n\nexact hn.symm",
"compiler_output_preview": "[path]:25:4: error: tactic 'apply' failed, could not unify the conclusion of `@rpow_pos_of_pos`\n 0 < ?x ^ ?y\nwith the goal\n 0 < \u03c0\n\nNote: The full type of `@rpow_pos_of_pos` is\n \u2200 {x : \u211d}, 0 < x \u2192 \u2200 (y : \u211d), 0 < x ^ y\ncase hx\nn : \u2124\nhn : \u03c0 ^ \u03c0 ^ \u03c0 ^ \u03c0 = \u2191n\nx : \u211d := \u22ef\nhx : x = \u2191n\n\u22a2 0 < \u03c0\n[path]
},
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"proof_code": "intro h\nrcases h with \u27e8n, hn\u27e9\nhave h_pi : 0 < \u03c0 := pi_pos\nhave h1 : 0 < \u03c0 ^ \u03c0 := rpow_pos_of_pos h_pi \u03c0\nhave h2 : 0 < \u03c0 ^ \u03c0 ^ \u03c0 := rpow_pos_of_pos h1 \u03c0\nhave h3 : 0 < \u03c0 ^ \u03c0 ^ \u03c0 ^ \u03c0 := rpow_pos_of_pos h2 \u03c0\nhave n_pos : 0 < n := by\n have : (0 : \u211d) < (n : \u211d) := by rwa [\u2190 hn]\n exact_mod_cast this\nlet x := \u03c0 ^ \u03c0 ^ \u03c0 ^ \u03c0\nsuffices : x \u2209 Set.range (Int.cast : \u2124 \u2192 \u211d)\n\u00b7 apply this\n use n\n exact hn.symm\nrefine ?_",
"compiler_output_preview": "[path]:23:29: error: type mismatch\n rpow_pos_of_pos h1 \u03c0\nhas type\n 0 < (\u03c0 ^ \u03c0) ^ \u03c0 : Prop\nbut is expected to have type\n 0 < \u03c0 ^ \u03c0 ^ \u03c0 : Prop\n[path]:24:33: error: type mismatch\n rpow_pos_of_pos h2 \u03c0\nhas type\n 0 < (\u03c0 ^ \u03c0 ^ \u03c0) ^\n...<truncated>"
},
{
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"banned_tokens": [],
"proof_code": "intro h\nrcases h with \u27e8n, hn\u27e9\nlet x := \u03c0 ^ (\u03c0 ^ (\u03c0 ^ \u03c0))\nhave h_pos : 0 < x := by\n apply rpow_pos_of_pos\n exact pi_pos\nhave n_pos : 0 < n := by\n have : (0 : \u211d) < (n : \u211d) := hn \u25b8 h_pos\n exact_mod_cast this\nhave h_not_int : x \u2209 Set.range (Int.cast : \u2124 \u2192 \u211d) := by\n intro (hx : \u2203 m : \u2124, x = m)\n refine ?_\napply h_not_int\nuse n\nexact hn.symm",
"compiler_output_preview": "[path]:29:4: error: type mismatch at `intro hx`has type\n x \u2208 Set.range Int.cast : Prop\nbut is expected to have type\n \u2203 m, x = \u2191m : Prop\n'Transcendental.pi_pow_pi_pow_pi_pow_pi_not_integer' depends on axioms: [propext, sorryAx, Classical.choice, Quot.sound]\n\n"
},
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"proof_code": "intro h\nrcases h with \u27e8n, hn\u27e9\nhave h_pi : 0 < \u03c0 := pi_pos\nhave h1 : 0 < \u03c0 ^ \u03c0 := rpow_pos_of_pos h_pi \u03c0\nhave h2 : 0 < \u03c0 ^ (\u03c0 ^ \u03c0) := rpow_pos_of_pos h_pi (\u03c0 ^ \u03c0)\nhave h3 : 0 < \u03c0 ^ (\u03c0 ^ (\u03c0 ^ \u03c0)) := rpow_pos_of_pos h_pi (\u03c0 ^ (\u03c0 ^ \u03c0))\nhave n_pos : 0 < n := by\n have : (0 : \u211d) < (n : \u211d) := by rw [\u2190 hn]; exact h3\n exact_mod_cast this\nhave h_trans : x \u2209 Set.range (Int.cast : \u2124 \u2192 \u211d) := by\n intro (hx : \u2203 m : \u2124, \u03c0 ^ (\u03c0 ^ (\u03c0 ^ \u03c0)) = m)\n refine ?_\napply h_trans\nuse n\nexact hn.symm",
"compiler_output_preview": "[path]:28:17: error: unknown identifier 'x'\n'Transcendental.pi_pow_pi_pow_pi_pow_pi_not_integer' depends on axioms: [propext, sorryAx, Classical.choice, Quot.sound]\n\n"
},
{
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"proof_code": "intro h\nrcases h with \u27e8n, hn\u27e9\nlet x := \u03c0 ^ (\u03c0 ^ (\u03c0 ^ \u03c0))\nhave hx : x = (n : \u211d) := hn\nhave h_pos : 0 < x := by\n apply rpow_pos_of_pos\n exact pi_pos\nhave n_pos : 0 < n := by\n have : (0 : \u211d) < (n : \u211d) := by rwa [\u2190 hn]\n exact_mod_cast this\nhave h_not_int : x \u2209 Set.range (Int.cast : \u2124 \u2192 \u211d) := by\n intro h_mem\n rcases h_mem with \u27e8m, hm\u27e9\n refine ?_\napply h_not_int\nuse n\nexact hn.symm",
"compiler_output_preview": "[path]:29:55: error: unsolved goals\ncase intro\nn : \u2124\nhn : \u03c0 ^ \u03c0 ^ \u03c0 ^ \u03c0 = \u2191n\nx : \u211d := \u03c0 ^ \u03c0 ^ \u03c0 ^ \u03c0\nhx : x = \u2191n\nh_pos : 0 < x\nn_pos : 0 < n\nm : \u2124\nhm : \u2191m = x\n\u22a2 False\n'Transcendental.pi_pow_pi_pow_pi_pow_pi_not_integer' depends on axioms: [propext, sorryAx, Classical.choice, Quot.sound]\n\n"
},
{
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"provider": "openrouter_primary",
"finish_reason": "stop",
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"usage": {
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{
"iteration": 8,
"event": "compile_proof",
"provider": "openrouter_primary",
"status": "failed",
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"banned_tokens": [],
"proof_code": "intro h\nrcases h with \u27e8n, hn\u27e9\nlet x := \u03c0 ^ (\u03c0 ^ (\u03c0 ^ \u03c0))\nhave h_pos : 0 < x := rpow_pos_of_pos pi_pos _\nhave n_pos : 0 < n := by\n have : (0 : \u211d) < (n : \u211d) := by rwa [\u2190 hn]\n exact_mod_cast this\nhave h_not_integer : x \u2209 Set.range (Int.cast : \u2124 \u2192 \u211d) := by\n intro (hx_mem : \u2203 m : \u2124, x = \u2191m)\n let val := x\n refine ?_\napply h_not_integer\nuse n\nexact hn.symm",
"compiler_output_preview": "[path]:27:4: error: type mismatch at `intro hx_mem`has type\n x \u2208 Set.range Int.cast : Prop\nbut is expected to have type\n \u2203 m, x = \u2191m : Prop\n'Transcendental.pi_pow_pi_pow_pi_pow_pi_not_integer' depends on axioms: [propext, sorryAx, Classical.choice, Quot.sound]\n\n"
}
],
"verification": {
"banned_tokens": [],
"command": [
"[path]
"lean",
"[path]
],
"duration_s": 7.162783622741699,
"goals_remaining": 1,
"is_verified": false,
"lean_file_path": "[path]
"problem": {
"axioms_check_name": "pi_pow_pi_pow_pi_pow_pi_not_integer",
"axioms_check_skipped": false,
"benchmark_id": "Wikipedia__Transcendental__pi_pow_pi_pow_pi_pow_pi_not_integer",
"category": "research_open",
"endings": [
"end Transcendental"
],
"imports": [
"FormalConjectures.Util.ProblemImports"
],
"preamble_chars": 167,
"source_file": "FormalConjectures/Wikipedia/Transcendental.lean",
"title": "pi_pow_pi_pow_pi_pow_pi_not_integer"
},
"returncode": 1,
"status": "partial",
"stderr": "",
"stdout": "[path]:29:55: error: unsolved goals\ncase intro\nn : \u2124\nhn : \u03c0 ^ \u03c0 ^ \u03c0 ^ \u03c0 = \u2191n\nx : \u211d := \u03c0 ^ \u03c0 ^ \u03c0 ^ \u03c0\nhx : x = \u2191n\nh_pos : 0 < x\nn_pos : 0 < n\nm : \u2124\nhm : \u2191m = x\n\u22a2 False\n'Transcendental.pi_pow_pi_pow_pi_pow_pi_not_integer' depends on axioms: [propext, sorryAx, Classical.choice, Quot.sound]\n",
"timed_out": false
}
}
|
| Gemini 3 Flash Preview | Failed | -- | Feb 14, 2026 |
have : 3.14 < π := Real.pi_gt_three.trans_le (by norm_num; linarith [Real.pi_gt_three])
linarith [Real.pi_gt_three]
have h_pow_pos : 0 < π ^ π ^ π ^ π := by
apply Real.rpow_pos_of_pos h_pi_pos
have h_n_pos : 0 < (n : ℝ) := by
rw [← hn]
exact h_pow_pos
have h_n_int : n > 0 := by
norm_cast at h_n_pos
exact h_n_pos
have h_trans : Transcendental π := transcendental_pi
have h_log : π ^ π ^ π ^ π = exp (π ^ π ^ π * log π) := by
rw [Real.rpow_def_of_pos h_pi_pos]
have h_eq : exp (π ^ π ^ π * log π) = n := by
rw [← h_log, hn]
apply Transcendental.transcendental_pi
apply Transcendental.isAlgebraic_iff.mp
rewrite [← hn] at *
contrapose! h_trans
exact transcendental_pi
|
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"stdout": "[path]:19:21: error: Function expected at\n LT.lt.trans_le pi_gt_three ?m.209\nbut this term has type\n 3 < ?m.196\n\nNote: Expected a function because this term is being applied to the argument\n linarith\n[path]:22:31: error: unknown identifier 'h_pi_pos'\n[path]:23:22: error: unknown identifier 'n'\n[path]:24:10: error: unknown identifier 'hn'\n[path]:26:17: error: unknown identifier 'n'\n[path]:29:32: error: Application type mismatch: In the application\n @Transcendental \u03c0\nthe argument\n \u03c0\nhas type\n \u211d : Type\nbut is expected to have type\n Type ?u.3061 : Type (?u.3061 + 1)\n[path]:29:37: error: unknown identifier 'transcendental_pi'\n[path]:31:29: error: unknown identifier 'h_pi_pos'\n[path]:32:40: error: unknown identifier 'n'\n[path]:33:17: error: unknown identifier 'hn'\n[path]:34:8: error: unknown constant 'Transcendental.transcendental_pi'\n[path]:35:2: error: no goals to be solved\n",
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|