56777.lean`

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

9 declarations, 1 contain sorry, 0 declared as axioms

Divisibility of $2^n + 1$ by $n$

A56777 lists composite numbers $n$ satisfying both $\varphi(n+12) = \varphi(n) + 12$ and $\sigma(n+12) = \sigma(n) + 12$.

The conjectures state identities connecting A56777 and prime quadruples (A7530), as well as congruences satisfied by the members of A56777.

References:

A56777

Imports (1)

FormalConjecturesUtil

Imported by 1 file

FormalConjectures/Subsets/FC100SolvedSet1.lean

def OeisA56777.A A composite number $n$ is in the sequence A56777 if it satisfies both $\varphi(n+12) = \varphi(n) + 12$ and $\sigma(n+12) = \sigma(n) + 12$ line 42 source
Source documentation
A composite number $n$ is in the sequence A56777 if it satisfies both $\varphi(n+12) = \varphi(n) + 12$ and $\sigma(n+12) = \sigma(n) + 12$.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat
Proof uses: And, Not, Nat.Prime, LT.lt, OfNat.ofNat, Eq, Nat.totient, HAdd.hAdd, ArithmeticFunction, ArithmeticFunction.sigma
8 instances, coercions and auxiliary constants left out.
```
def A (n : ℕ) : Prop :=
…
```
def OeisA56777.ComesFromPrimeQuadruple A number $n$ comes from a prime quadruple $(p, p+2, p+6, p+8)$ if $n = p(p+8)$ for some prime $p$ where $p$, $p+2$, $p+6$, $p+8$ are all prime line 47 source
Source documentation
A number $n$ comes from a prime quadruple $(p, p+2, p+6, p+8)$ if $n = p(p+8)$ for some prime $p$ where $p$, $p+2$, $p+6$, $p+8$ are all prime.

standard axioms: propext what this means
Statement uses: Nat
Proof uses: Exists, And, Nat.Prime, HAdd.hAdd, OfNat.ofNat, Eq, HMul.hMul
5 instances, coercions and auxiliary constants left out.
```
def ComesFromPrimeQuadruple (n : ℕ) : Prop :=
…
```
theorem OeisA56777.a_65 $65$ is in the sequence A56777 line 52 source
Source documentation
$65$ is in the sequence A56777.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: OeisA56777.A, OfNat.ofNat, Nat
Proof uses: And.intro, Not, Nat.Prime, And, LT.lt, Eq, Nat.totient, HAdd.hAdd, ArithmeticFunction, ArithmeticFunction.sigma, Eq.mpr, HMul.hMul, id, congrArg, Mathlib.Meta.NormNum.isNat_eq_true, Mathlib.Meta.NormNum.isNat_ofNat, Eq.refl, Mathlib.Meta.NormNum.isNat_mul, Nat.not_prime_mul, Mathlib.Meta.NormNum.isNat_eq_false and 6 more
22 instances, coercions and auxiliary constants left out.

Tactic trace: 12 steps
Open to load the checked before-and-after goal states.
```
theorem a_65 : A 65 := by
…
```
Mentions: OeisA56777.A
theorem OeisA56777.a_209 $209$ is in the sequence A56777 line 61 source
Source documentation
$209$ is in the sequence A56777.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: OeisA56777.A, OfNat.ofNat, Nat
Proof uses: id, Eq.mpr, And, Not, Nat.Prime, LT.lt, Eq, Nat.totient, HAdd.hAdd, ArithmeticFunction, ArithmeticFunction.sigma, congrArg, Eq.trans, True, congrFun', true_and, And.intro, Mathlib.Meta.NormNum.isNat_not_prime, Mathlib.Meta.NormNum.isNat_ofNat, Eq.refl and 20 more
28 instances, coercions and auxiliary constants left out.

Tactic trace: 19 steps
Open to load the checked before-and-after goal states.
```
theorem a_209 : A 209 := by
…
```
Mentions: OeisA56777.A
theorem OeisA56777.a_of_comesFromPrimeQuadruple Numbers coming from prime quadruples are in the sequence A56777 line 75 source
Source documentation
Numbers coming from prime quadruples are in the sequence A56777.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, OeisA56777.ComesFromPrimeQuadruple, OeisA56777.A
Proof uses: And, Nat.Prime, HAdd.hAdd, OfNat.ofNat, Eq, HMul.hMul, Mathlib.Tactic.Ring.of_eq, Nat.rawCast, HPow.hPow, Mathlib.Tactic.Ring.Common.add_congr, Mathlib.Tactic.Ring.Common.mul_congr, Mathlib.Tactic.Ring.Common.atom_pf, rfl, Eq.mpr, id, congrArg, Eq.symm, Eq.refl, Mathlib.Tactic.Ring.cast_pos, Mathlib.Meta.NormNum.isNat_ofNat and 89 more
119 instances, coercions and auxiliary constants left out.

Tactic trace: 42 steps
Open to load the checked before-and-after goal states.
```
theorem a_of_comesFromPrimeQuadruple {n : ℕ} (h : ComesFromPrimeQuadruple n) : A n := by
…
```
Mentions: OeisA56777.ComesFromPrimeQuadruple, OeisA56777.A
theorem OeisA56777.a_11009 $11009$ is in the sequence A56777 line 118 source
Source documentation
$11009$ is in the sequence A56777.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: OeisA56777.A, OfNat.ofNat, Nat
Proof uses: OeisA56777.a_of_comesFromPrimeQuadruple, Exists.intro, And, Nat.Prime, HAdd.hAdd, Eq, HMul.hMul, And.intro, of_decide_eq_true, id, Bool, Decidable.decide, Bool.true, Eq.refl
7 instances, coercions and auxiliary constants left out.

Tactic trace: 7 steps
Open to load the checked before-and-after goal states.
```
theorem a_11009 : A 11009 := by
…
```
Mentions: OeisA56777.A
theorem OeisA56777.comesFromPrimeQuadruple_of_a All members of the sequence A56777 come from prime quadruples sorry line 124 source
Source documentation
All members of the sequence A56777 come from prime quadruples.

depends on sorry what this means
Statement uses: Nat, OeisA56777.A, OeisA56777.ComesFromPrimeQuadruple
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous, OfNat.ofNat
1 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem comesFromPrimeQuadruple_of_a {n : ℕ} (h : A n) : ComesFromPrimeQuadruple n := by
…
```
Mentions: OeisA56777.A, OeisA56777.ComesFromPrimeQuadruple
theorem OeisA56777.mod_72_of_comesFromPrimeQuadruple Numbers coming from prime quadruples satisfy $n \equiv 65 \pmod{72}$ line 129 source
Source documentation
Numbers coming from prime quadruples satisfy $n \equiv 65 \pmod{72}$.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, OeisA56777.ComesFromPrimeQuadruple, Eq, HMod.hMod, OfNat.ofNat
Proof uses: And, Nat.Prime, HAdd.hAdd, HMul.hMul, LE.le, Decidable.byContradiction, Not, dite, False, LT.lt, False.elim, Eq.mp, eq_false_of_decide, Eq.refl, Bool, Bool.false, Eq.symm, le_antisymm, Nat.ge_of_not_lt, Mathlib.Tactic.IntervalCases.of_lt_right and 89 more
122 instances, coercions and auxiliary constants left out.

Tactic trace: 50 steps
Open to load the checked before-and-after goal states.
```
theorem mod_72_of_comesFromPrimeQuadruple {n : ℕ} (h : ComesFromPrimeQuadruple n) :
    n % 72 = 65 := by
…
```
Mentions: OeisA56777.ComesFromPrimeQuadruple
theorem OeisA56777.mod_100_of_comesFromPrimeQuadruple Numbers coming from prime quadruples satisfy $n \equiv 9 \pmod{100}$, except the first value "65" line 163 source
Source documentation
Numbers coming from prime quadruples satisfy $n \equiv 9 \pmod{100}$, except the first value "65".

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, LT.lt, OfNat.ofNat, OeisA56777.ComesFromPrimeQuadruple, Eq, HMod.hMod
Proof uses: And, Nat.Prime, HAdd.hAdd, HMul.hMul, LE.le, Decidable.byContradiction, Not, dite, False, False.elim, Eq.mp, eq_false_of_decide, Eq.refl, Bool, Bool.false, Eq.symm, le_antisymm, Nat.ge_of_not_lt, Mathlib.Tactic.IntervalCases.of_lt_right, Nat.rawCast and 89 more
133 instances, coercions and auxiliary constants left out.

Tactic trace: 80 steps
Open to load the checked before-and-after goal states.
```
theorem mod_100_of_comesFromPrimeQuadruple {n : ℕ} (h65 : 65 < n) (h : ComesFromPrimeQuadruple n) :
    n % 100 = 9 := by
…
```
Mentions: OeisA56777.ComesFromPrimeQuadruple