232174.lean`

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

8 declarations, 1 contain sorry, 0 declared as axioms

Representations with prime conditions

Any integer $n > 1$ can be written as $x + y$ with $x, y > 0$ such that both $x + ny$ and $x^2 + ny^2$ are prime.

Zhi-Wei Sun has offered a $200 prize for the first proof.

References:

A232174
Z.-W. Sun, "Conjectures on representations involving primes," in: M. Nathanson (ed.), Combinatorial and Additive Number Theory II: CANT, Springer Proc. in Math. & Stat., Vol. 220, Springer, 2017, pp. 279-310. https://arxiv.org/abs/1211.1588
D.A. Cox, "Primes of the Form x² + ny²," John Wiley & Sons, 1989.

Imports (1)

FormalConjecturesUtil

Imported by 1 file

FormalConjectures/Subsets/FC100SolvedSet1.lean

def OeisA232174.A The predicate that n can be written as $x + y$ with $x, y > 0$ such that both $x + ny$ and $x^2 + ny^2$ are prime line 42 source
Source documentation
The predicate that n can be written as $x + y$ with $x, y > 0$ such that both $x + ny$ and $x^2 + ny^2$ are prime.

standard axioms: propext what this means
Statement uses: Nat
Proof uses: Exists, And, LT.lt, OfNat.ofNat, Eq, HAdd.hAdd, Nat.Prime, HMul.hMul, HPow.hPow
10 instances, coercions and auxiliary constants left out.
```
def A (n : ℕ) : Prop :=
…
```
theorem OeisA232174.a_2 line 46 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: OeisA232174.A, OfNat.ofNat, Nat
Proof uses: Exists.intro, Exists, And, LT.lt, Eq, HAdd.hAdd, Nat.Prime, HMul.hMul, 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_mul, Bool.true and 5 more
24 instances, coercions and auxiliary constants left out.

Tactic trace: 5 steps
Open to load the checked before-and-after goal states.
theorem OeisA232174.a_3 line 50 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: OeisA232174.A, OfNat.ofNat, Nat
Proof uses: Exists.intro, Exists, And, LT.lt, Eq, HAdd.hAdd, Nat.Prime, HMul.hMul, 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_mul, Bool.true and 7 more
24 instances, coercions and auxiliary constants left out.

Tactic trace: 5 steps
Open to load the checked before-and-after goal states.
theorem OeisA232174.a_4 line 54 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: OeisA232174.A, OfNat.ofNat, Nat
Proof uses: Exists.intro, Exists, And, LT.lt, Eq, HAdd.hAdd, Nat.Prime, HMul.hMul, 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_mul, Bool.true and 8 more
24 instances, coercions and auxiliary constants left out.

Tactic trace: 5 steps
Open to load the checked before-and-after goal states.
theorem OeisA232174.a_5 line 58 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: OeisA232174.A, OfNat.ofNat, Nat
Proof uses: Exists.intro, Exists, And, LT.lt, Eq, HAdd.hAdd, Nat.Prime, HMul.hMul, 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_mul, Bool.true and 7 more
24 instances, coercions and auxiliary constants left out.

Tactic trace: 5 steps
Open to load the checked before-and-after goal states.
theorem OeisA232174.a_6 line 62 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: OeisA232174.A, OfNat.ofNat, Nat
Proof uses: Exists.intro, Exists, And, LT.lt, Eq, HAdd.hAdd, Nat.Prime, HMul.hMul, 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_mul, Bool.true and 8 more
24 instances, coercions and auxiliary constants left out.

Tactic trace: 5 steps
Open to load the checked before-and-after goal states.
theorem OeisA232174.a_8 line 66 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: OeisA232174.A, OfNat.ofNat, Nat
Proof uses: Exists.intro, Exists, And, LT.lt, Eq, HAdd.hAdd, Nat.Prime, HMul.hMul, 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_mul, Bool.true and 9 more
24 instances, coercions and auxiliary constants left out.

Tactic trace: 5 steps
Open to load the checked before-and-after goal states.
theorem OeisA232174.conjecture Zhi-Wei Sun's Conjecture (A232174): Any integer $n > 1$ can be written as $x + y$ with $x, y > 0$ such that both $x + ny$ and $x^2 + ny^2$ are prime sorry line 74 source
Source documentation
Zhi-Wei Sun's Conjecture (A232174): Any integer $n > 1$ can be written as $x + y$ with $x, y > 0$ such that both $x + ny$ and $x^2 + ny^2$ are prime.

depends on sorry what this means
Statement uses: Nat, LT.lt, OfNat.ofNat, OeisA232174.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: OeisA232174.A