248802.lean`

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Project sources: Pinned project documentation, Formal Conjectures paper, arXiv v1

11 declarations, 2 contain sorry, 0 declared as axioms

Smallest prime factors of $2^{2^n+2} + 3$

Smallest prime factor of $2^{2^n+2} + 3$.

References:

A248802
arxiv/2605.22763 Advancing Mathematics Research with AI-Driven Formal Proof Search by George Tsoukalas et al.

Imports (1)

FormalConjecturesUtil

def OeisA248802.a Smallest prime factor of $2^{2^n+2} + 3$ line 38 source
Source documentation
Smallest prime factor of $2^{2^n+2} + 3$.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat
Proof uses: Nat.minFac, HAdd.hAdd, HPow.hPow, OfNat.ofNat
7 instances, coercions and auxiliary constants left out.
```
def a (n : ℕ) : ℕ := (2 ^ (2 ^ n + 2) + 3).minFac
```
def OeisA248802.CoveredByC1 An index k is covered by Conjecture 1 if k = 10m + 2 for some m >= 0, predicting a(k)=67 line 41 source
Source documentation
An index k is covered by Conjecture 1 if k = 10m + 2 for some m >= 0, predicting a(k)=67.

no axioms what this means
Statement uses: Nat
Proof uses: Exists, Eq, HAdd.hAdd, HMul.hMul, OfNat.ofNat
5 instances, coercions and auxiliary constants left out.
```
def CoveredByC1 (k : ℕ) : Prop := ∃ m : ℕ, k = 10 * m + 2
```
def OeisA248802.CoveredByC2 An index k is covered by Conjecture 2 if k = 36m + 16 for some m >= 0, and m is not 1 mod 5, predicting a(k)=271 line 45 source
Source documentation
An index k is covered by Conjecture 2 if k = 36m + 16 for some m >= 0, and m is not 1 mod 5, predicting a(k)=271.

no axioms what this means
Statement uses: Nat
Proof uses: Exists, And, Eq, HAdd.hAdd, HMul.hMul, OfNat.ofNat, Ne, HMod.hMod
7 instances, coercions and auxiliary constants left out.
```
def CoveredByC2 (k : ℕ) : Prop := ∃ m : ℕ, k = 36 * m + 16 ∧ m % 5 ≠ 1
```
def OeisA248802.CoveredByC3 An index k is covered by Conjecture 3 if k = 84m + 22 for some m >= 0, and m is not 0 mod 5, predicting a(k)=523 line 49 source
Source documentation
An index k is covered by Conjecture 3 if k = 84m + 22 for some m >= 0, and m is not 0 mod 5, predicting a(k)=523.

no axioms what this means
Statement uses: Nat
Proof uses: Exists, And, Eq, HAdd.hAdd, HMul.hMul, OfNat.ofNat, Ne, HMod.hMod
7 instances, coercions and auxiliary constants left out.
```
def CoveredByC3 (k : ℕ) : Prop := ∃ m : ℕ, k = 84 * m + 22 ∧ m % 5 ≠ 0
…
```
lemma OeisA248802.a_0 line 53 source
extra axioms: _private.0.OeisA248802.a_0._native.native_decide.ax_1 what this means
Statement uses: Eq, Nat, OeisA248802.a, OfNat.ofNat
Proof uses: of_decide_eq_true
4 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
lemma OeisA248802.a_1 line 56 source
extra axioms: _private.0.OeisA248802.a_1._native.native_decide.ax_1 what this means
Statement uses: Eq, Nat, OeisA248802.a, OfNat.ofNat
Proof uses: of_decide_eq_true
4 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
lemma OeisA248802.a_2 line 59 source
extra axioms: _private.0.OeisA248802.a_2._native.native_decide.ax_1 what this means
Statement uses: Eq, Nat, OeisA248802.a, OfNat.ofNat
Proof uses: of_decide_eq_true
4 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
lemma OeisA248802.a_3 line 62 source
extra axioms: _private.0.OeisA248802.a_3._native.native_decide.ax_1 what this means
Statement uses: Eq, Nat, OeisA248802.a, OfNat.ofNat
Proof uses: of_decide_eq_true
4 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
lemma OeisA248802.a_4 line 65 source
extra axioms: _private.0.OeisA248802.a_4._native.native_decide.ax_1 what this means
Statement uses: Eq, Nat, OeisA248802.a, OfNat.ofNat
Proof uses: of_decide_eq_true
4 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
theorem OeisA248802.a_ten_mul_add_two_eq sorry line 76 source
depends on sorry what this means
Statement uses: Nat, Eq, OeisA248802.a, HAdd.hAdd, HMul.hMul, OfNat.ofNat
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
10 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
theorem OeisA248802.a_fifty_eight_mul_add_twenty_six_eq sorry line 88 source
depends on sorry what this means
Statement uses: Nat, And, Not, OeisA248802.CoveredByC1, HAdd.hAdd, HMul.hMul, OfNat.ofNat, OeisA248802.CoveredByC2, OeisA248802.CoveredByC3, Eq, OeisA248802.a
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
10 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.