231201.lean`

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

8 declarations, 1 contain sorry, 0 declared as axioms

Sum of two numbers with prime conditions

Number of ways to write $n = x+y$, for $x,y > 0$ such that $2^x + y$ is prime.

Zhi-Wei Sun has offered a $1000 prize for the first proof.

References:

A231201
Zhi-Wei Sun, "Table of n, a(n) for n = 1..10000", "Write n = k + m with 2^k + m prime", a message to Number Theory List, Nov. 16, 2013, "On a^n+ bn modulo m", arXiv:1312.1166 [math.NT], 2013-2014, "Problems on combinatorial properties of primes", arXiv:1402.6641 [math.NT], 2014-2015.

Imports (1)

FormalConjecturesUtil

Imported by 1 file

FormalConjectures/Subsets/FC100SolvedSet1.lean

def OeisA231201.A The predicate that n can be written as $x + y$ with $x,y >0$ such that $2^x + y$ is prime line 41 source
Source documentation
The predicate that n can be written as $x + y$ with $x,y >0$ such that $2^x + y$ is prime

standard axioms: propext what this means
Statement uses: Nat
Proof uses: Exists, And, LT.lt, OfNat.ofNat, Eq, HAdd.hAdd, Nat.Prime, HPow.hPow
8 instances, coercions and auxiliary constants left out.
```
def A (n : ℕ) : Prop :=
…
```
theorem OeisA231201.a_2 line 45 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: OeisA231201.A, OfNat.ofNat, Nat
Proof uses: Exists.intro, Exists, And, LT.lt, Eq, HAdd.hAdd, Nat.Prime, HPow.hPow, And.intro, Mathlib.Meta.NormNum.isNat_lt_true, Mathlib.Meta.NormNum.isNat_ofNat, Eq.refl, Bool, Bool.false, Mathlib.Meta.NormNum.isNat_eq_true, Mathlib.Meta.NormNum.isNat_add, Mathlib.Meta.NormNum.isNat_prime_2, Mathlib.Meta.NormNum.isNat_pow, Mathlib.Meta.NormNum.natPow_one, Bool.true and 3 more
20 instances, coercions and auxiliary constants left out.

Tactic trace: 4 steps
Open to load the checked before-and-after goal states.
theorem OeisA231201.a_3 line 49 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: OeisA231201.A, OfNat.ofNat, Nat
Proof uses: Exists.intro, Exists, And, LT.lt, Eq, HAdd.hAdd, Nat.Prime, HPow.hPow, And.intro, Mathlib.Meta.NormNum.isNat_lt_true, Mathlib.Meta.NormNum.isNat_ofNat, Eq.refl, Bool, Bool.false, Mathlib.Meta.NormNum.isNat_eq_true, Mathlib.Meta.NormNum.isNat_add, Mathlib.Meta.NormNum.isNat_prime_2, Mathlib.Meta.NormNum.isNat_pow, Mathlib.Meta.NormNum.IsNatPowT.run, Mathlib.Meta.NormNum.IsNatPowT.bit0 and 4 more
20 instances, coercions and auxiliary constants left out.

Tactic trace: 4 steps
Open to load the checked before-and-after goal states.
theorem OeisA231201.a_4 line 53 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: OeisA231201.A, OfNat.ofNat, Nat
Proof uses: Exists.intro, Exists, And, LT.lt, Eq, HAdd.hAdd, Nat.Prime, HPow.hPow, And.intro, Mathlib.Meta.NormNum.isNat_lt_true, Mathlib.Meta.NormNum.isNat_ofNat, Eq.refl, Bool, Bool.false, Mathlib.Meta.NormNum.isNat_eq_true, Mathlib.Meta.NormNum.isNat_add, Mathlib.Meta.NormNum.isNat_prime_2, Mathlib.Meta.NormNum.isNat_pow, Mathlib.Meta.NormNum.natPow_one, Bool.true and 3 more
20 instances, coercions and auxiliary constants left out.

Tactic trace: 4 steps
Open to load the checked before-and-after goal states.
theorem OeisA231201.a_5 line 57 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: OeisA231201.A, OfNat.ofNat, Nat
Proof uses: Exists.intro, Exists, And, LT.lt, Eq, HAdd.hAdd, Nat.Prime, HPow.hPow, And.intro, Mathlib.Meta.NormNum.isNat_lt_true, Mathlib.Meta.NormNum.isNat_ofNat, Eq.refl, Bool, Bool.false, Mathlib.Meta.NormNum.isNat_eq_true, Mathlib.Meta.NormNum.isNat_add, Mathlib.Meta.NormNum.isNat_prime_2, Mathlib.Meta.NormNum.isNat_pow, Mathlib.Meta.NormNum.IsNatPowT.run, Mathlib.Meta.NormNum.IsNatPowT.bit0 and 4 more
20 instances, coercions and auxiliary constants left out.

Tactic trace: 4 steps
Open to load the checked before-and-after goal states.
theorem OeisA231201.a_8 line 61 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: OeisA231201.A, OfNat.ofNat, Nat
Proof uses: Exists.intro, Exists, And, LT.lt, Eq, HAdd.hAdd, Nat.Prime, HPow.hPow, And.intro, Mathlib.Meta.NormNum.isNat_lt_true, Mathlib.Meta.NormNum.isNat_ofNat, Eq.refl, Bool, Bool.false, Mathlib.Meta.NormNum.isNat_eq_true, Mathlib.Meta.NormNum.isNat_add, Mathlib.Meta.NormNum.isNat_prime_2, Mathlib.Meta.NormNum.isNat_pow, Mathlib.Meta.NormNum.IsNatPowT.run, Mathlib.Meta.NormNum.IsNatPowT.bit1 and 5 more
20 instances, coercions and auxiliary constants left out.

Tactic trace: 4 steps
Open to load the checked before-and-after goal states.
theorem OeisA231201.a_53 line 65 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: OeisA231201.A, OfNat.ofNat, Nat
Proof uses: Exists.intro, Exists, And, LT.lt, Eq, HAdd.hAdd, Nat.Prime, HPow.hPow, And.intro, Mathlib.Meta.NormNum.isNat_lt_true, Mathlib.Meta.NormNum.isNat_ofNat, Eq.refl, Bool, Bool.false, Mathlib.Meta.NormNum.isNat_eq_true, Mathlib.Meta.NormNum.isNat_add, Mathlib.Meta.NormNum.isNat_prime_2, Mathlib.Meta.NormNum.isNat_pow, Mathlib.Meta.NormNum.IsNatPowT.run, Mathlib.Meta.NormNum.IsNatPowT.trans and 10 more
20 instances, coercions and auxiliary constants left out.

Tactic trace: 4 steps
Open to load the checked before-and-after goal states.
theorem OeisA231201.conjecture The conjecture for sequence A231201: for any $n > 1$, there exist $x, y > 0$ such that $n = x + y$ and $2^x + y$ is prime sorry line 70 source
Source documentation
The conjecture for sequence A231201: for any $n > 1$, there exist $x, y > 0$ such that $n = x + y$ and $2^x + y$ is prime.

depends on sorry what this means
Statement uses: Nat, LT.lt, OfNat.ofNat, OeisA231201.A
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 conjecture (n : ℕ) (hn : 1 < n) : A n := by
…
```
Mentions: OeisA231201.A