37274.lean`

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

10 declarations, 1 contain sorry, 0 declared as axioms

Home primes (OEIS A037274)

Starting from an integer $n\geq 2$, list its prime factors in nondecreasing order with multiplicity, concatenate their decimal representations, and repeat. The home-prime conjecture says that this process always reaches a prime.

For example,

$$25 \longmapsto 55 \longmapsto 511 \longmapsto 773.$$

References:

OEIS A037274
M. Herman and J. Schiffman, Investigating home primes and their families, Mathematics Teacher 107 (2014), 606–614

Imports (1)

FormalConjecturesUtil

def OeisA37274.decimalDigitCount The number of decimal digits of a natural number, counting zero as one digit line 42 source
Source documentation
The number of decimal digits of a natural number, counting zero as one digit.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat
Proof uses: ite, Eq, OfNat.ofNat, List.length, Nat.digits
2 instances, coercions and auxiliary constants left out.
```
def decimalDigitCount (n : ℕ) : ℕ :=
…
```
def OeisA37274.decimalAppend Append the decimal digits of b to those of a line 46 source
Source documentation
Append the decimal digits of b to those of a.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat
Proof uses: HAdd.hAdd, HMul.hMul, HPow.hPow, OfNat.ofNat, OeisA37274.decimalDigitCount
9 instances, coercions and auxiliary constants left out.
```
def decimalAppend (a b : ℕ) : ℕ :=
…
```
def OeisA37274.primeFactorSplice Concatenate the prime factors of n in nondecreasing order, retaining multiplicity line 50 source
Source documentation
Concatenate the prime factors of n in nondecreasing order, retaining multiplicity.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat
Proof uses: List.foldl, OeisA37274.decimalAppend, OfNat.ofNat, Nat.primeFactorsList
1 instances, coercions and auxiliary constants left out.
```
def primeFactorSplice (n : ℕ) : ℕ :=
…
```
def OeisA37274.ReachesPrime A starting value reaches a prime after finitely many prime-factor splicing steps line 54 source
Source documentation
A starting value reaches a prime after finitely many prime-factor splicing steps.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat
Proof uses: Exists, Nat.Prime, Nat.iterate, OeisA37274.primeFactorSplice
```
def ReachesPrime (n : ℕ) : Prop :=
…
```
theorem OeisA37274.home_prime_conjecture Every integer at least two reaches a home prime sorry line 59 source
Source documentation
Every integer at least two reaches a home prime.

depends on sorry what this means
Statement uses: Nat, LE.le, OfNat.ofNat, OeisA37274.ReachesPrime
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
4 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem home_prime_conjecture : ∀ n : ℕ, 2 ≤ n → ReachesPrime n := by
…
```
Mentions: OeisA37274.ReachesPrime
theorem OeisA37274.primeFactorSplice_25 The first step in the trajectory from $25$ is $25\mapsto55$ line 64 source
Source documentation
The first step in the trajectory from $25$ is $25\mapsto55$.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA37274.primeFactorSplice, OfNat.ofNat
Proof uses: of_eq_true, Eq.trans, True, congrFun', congrArg, List.foldl, OeisA37274.decimalAppend, Nat.primeFactorsList, List.cons, List.nil, List, Nat.succ, Nat.minFac, HAdd.hAdd, HDiv.hDiv, Nat.primeFactorsList.eq_3, congr, Mathlib.Meta.NormNum.IsNat.to_eq, Mathlib.Meta.NormNum.isNat_minFac_3, Mathlib.Meta.NormNum.isNat_add and 35 more
34 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem primeFactorSplice_25 : primeFactorSplice 25 = 55 := by
…
```
Mentions: OeisA37274.primeFactorSplice
theorem OeisA37274.primeFactorSplice_55 The second step in the trajectory from $25$ is $55\mapsto511$ line 69 source
Source documentation
The second step in the trajectory from $25$ is $55\mapsto511$.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA37274.primeFactorSplice, OfNat.ofNat
Proof uses: of_eq_true, Eq.trans, True, congrFun', congrArg, List.foldl, OeisA37274.decimalAppend, Nat.primeFactorsList, List.cons, List.nil, List, Nat.succ, Nat.minFac, HAdd.hAdd, HDiv.hDiv, Nat.primeFactorsList.eq_3, congr, Mathlib.Meta.NormNum.IsNat.to_eq, Mathlib.Meta.NormNum.isNat_minFac_3, Mathlib.Meta.NormNum.isNat_add and 38 more
34 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem primeFactorSplice_55 : primeFactorSplice 55 = 511 := by
…
```
Mentions: OeisA37274.primeFactorSplice
theorem OeisA37274.primeFactorSplice_511 The third step in the trajectory from $25$ is $511\mapsto773$ line 74 source
Source documentation
The third step in the trajectory from $25$ is $511\mapsto773$.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA37274.primeFactorSplice, OfNat.ofNat
Proof uses: of_eq_true, Eq.trans, True, congrFun', congrArg, List.foldl, OeisA37274.decimalAppend, Nat.primeFactorsList, List.cons, List.nil, List, Nat.succ, Nat.minFac, HAdd.hAdd, HDiv.hDiv, Nat.primeFactorsList.eq_3, congr, Mathlib.Meta.NormNum.IsNat.to_eq, Mathlib.Meta.NormNum.isNat_minFac_3, Mathlib.Meta.NormNum.isNat_add and 38 more
34 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem primeFactorSplice_511 : primeFactorSplice 511 = 773 := by
…
```
Mentions: OeisA37274.primeFactorSplice
theorem OeisA37274.primeFactorSplice_prime A prime is a fixed point of prime-factor splicing line 79 source
Source documentation
A prime is a fixed point of prime-factor splicing.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Nat.Prime, Eq, OeisA37274.primeFactorSplice
Proof uses: of_eq_true, Eq.trans, True, congrFun', congrArg, List.foldl, OeisA37274.decimalAppend, OfNat.ofNat, Nat.primeFactorsList, List.cons, List.nil, List, Nat.primeFactorsList_prime, HAdd.hAdd, HMul.hMul, HPow.hPow, OeisA37274.decimalDigitCount, MulZeroClass.zero_mul, zero_add, eq_self
14 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem primeFactorSplice_prime {p : ℕ} (hp : p.Prime) : primeFactorSplice p = p := by
…
```
Mentions: OeisA37274.primeFactorSplice
theorem OeisA37274.reachesPrime_25 The trajectory from $25$ reaches the prime $773$ after three steps line 84 source
Source documentation
The trajectory from $25$ reaches the prime $773$ after three steps.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: OeisA37274.ReachesPrime, OfNat.ofNat, Nat
Proof uses: Exists.intro, Nat.Prime, Nat.iterate, OeisA37274.primeFactorSplice, of_eq_true, Eq.trans, True, congrArg, List.foldl, OeisA37274.decimalAppend, List.nil, Nat.primeFactorsList, List.cons, List, Nat.succ, Nat.minFac, HAdd.hAdd, HDiv.hDiv, Nat.primeFactorsList.eq_3, congr and 47 more
34 instances, coercions and auxiliary constants left out.

Tactic trace: 2 steps
Open to load the checked before-and-after goal states.
```
theorem reachesPrime_25 : ReachesPrime 25 := by
…
```
Mentions: OeisA37274.ReachesPrime