303656.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 squares, a power of 3, and a power of 5

Any integer $n > 1$ can be written as $a^2 + b^2 + 3^c + 5^d$ where $a, b, c, d$ are nonnegative integers.

Zhi-Wei Sun has offered a $3,500 prize for the first proof.

References:

A303656
Z.-W. Sun, "Restricted sums of four squares," arXiv preprint: https://arxiv.org/abs/1701.05868v10
Z.-W. Sun, "Refining Lagrange's four-square theorem," Journal of Number Theory: http://maths.nju.edu.cn/~zwsun/RefineFourSquareTh.pdf
Z.-W. Sun, "Restricted sums of three or four squares": http://maths.nju.edu.cn/~zwsun/Square-sum.pdf
Zhi-Wei Sun's 1-3-5 conjecture and variations: https://www.aimspress.com/aimspress-data/era/2020/2/PDF/1935-9179_2020_2_589.pdf

Imports (1)

FormalConjecturesUtil

Imported by 1 file

FormalConjectures/Subsets/FC100OpenSet1.lean

def OeisA303656.A The predicate that n can be written as $a^2 + b^2 + 3^c + 5^d$ for nonnegative integers line 45 source
Source documentation
The predicate that n can be written as $a^2 + b^2 + 3^c + 5^d$ for nonnegative integers.

standard axioms: propext what this means
Statement uses: Nat
Proof uses: Exists, Eq, HAdd.hAdd, HPow.hPow, OfNat.ofNat
7 instances, coercions and auxiliary constants left out.
```
def A (n : ℕ) : Prop :=
…
```
theorem OeisA303656.a_2 line 49 source
standard axioms: propext what this means
Statement uses: OeisA303656.A, OfNat.ofNat, Nat
Proof uses: Exists.intro, Exists, Eq, HAdd.hAdd, HPow.hPow, Mathlib.Meta.NormNum.isNat_eq_true, Mathlib.Meta.NormNum.isNat_ofNat, Eq.refl, Mathlib.Meta.NormNum.isNat_add, Mathlib.Meta.NormNum.isNat_pow, Mathlib.Meta.NormNum.zero_natPow, Mathlib.Meta.NormNum.natPow_zero
13 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
theorem OeisA303656.a_3 line 53 source
standard axioms: propext what this means
Statement uses: OeisA303656.A, OfNat.ofNat, Nat
Proof uses: Exists.intro, Exists, Eq, HAdd.hAdd, HPow.hPow, Mathlib.Meta.NormNum.isNat_eq_true, Mathlib.Meta.NormNum.isNat_ofNat, Eq.refl, Mathlib.Meta.NormNum.isNat_add, Mathlib.Meta.NormNum.isNat_pow, Mathlib.Meta.NormNum.one_natPow, Mathlib.Meta.NormNum.zero_natPow, Mathlib.Meta.NormNum.natPow_zero
13 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
theorem OeisA303656.a_4 line 57 source
standard axioms: propext what this means
Statement uses: OeisA303656.A, OfNat.ofNat, Nat
Proof uses: Exists.intro, Exists, Eq, HAdd.hAdd, HPow.hPow, Mathlib.Meta.NormNum.isNat_eq_true, Mathlib.Meta.NormNum.isNat_ofNat, Eq.refl, Mathlib.Meta.NormNum.isNat_add, Mathlib.Meta.NormNum.isNat_pow, Mathlib.Meta.NormNum.one_natPow, Mathlib.Meta.NormNum.natPow_zero
13 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
theorem OeisA303656.a_5 line 61 source
standard axioms: propext what this means
Statement uses: OeisA303656.A, OfNat.ofNat, Nat
Proof uses: Exists.intro, Exists, Eq, HAdd.hAdd, HPow.hPow, Mathlib.Meta.NormNum.isNat_eq_true, Mathlib.Meta.NormNum.isNat_ofNat, Eq.refl, Mathlib.Meta.NormNum.isNat_add, Mathlib.Meta.NormNum.isNat_pow, Mathlib.Meta.NormNum.zero_natPow, Mathlib.Meta.NormNum.one_natPow, Mathlib.Meta.NormNum.natPow_one, Mathlib.Meta.NormNum.natPow_zero
13 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
theorem OeisA303656.a_6 line 65 source
standard axioms: propext what this means
Statement uses: OeisA303656.A, OfNat.ofNat, Nat
Proof uses: Exists.intro, Exists, Eq, HAdd.hAdd, HPow.hPow, Mathlib.Meta.NormNum.isNat_eq_true, Mathlib.Meta.NormNum.isNat_ofNat, Eq.refl, Mathlib.Meta.NormNum.isNat_add, Mathlib.Meta.NormNum.isNat_pow, Mathlib.Meta.NormNum.zero_natPow, Mathlib.Meta.NormNum.natPow_zero, Mathlib.Meta.NormNum.natPow_one
13 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
theorem OeisA303656.a_25 line 69 source
standard axioms: propext what this means
Statement uses: OeisA303656.A, OfNat.ofNat, Nat
Proof uses: Exists.intro, Exists, Eq, HAdd.hAdd, HPow.hPow, Mathlib.Meta.NormNum.isNat_eq_true, Mathlib.Meta.NormNum.isNat_ofNat, Eq.refl, Mathlib.Meta.NormNum.isNat_add, Mathlib.Meta.NormNum.isNat_pow, Mathlib.Meta.NormNum.one_natPow, Mathlib.Meta.NormNum.IsNatPowT.run, Mathlib.Meta.NormNum.IsNatPowT.bit0, Mathlib.Meta.NormNum.natPow_one
13 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
theorem OeisA303656.conjecture Zhi-Wei Sun's Conjecture (A303656): Any integer $n > 1$ can be written as the sum of two squares, a power of 3, and a power of 5 sorry line 77 source
Source documentation
Zhi-Wei Sun's Conjecture (A303656): Any integer $n > 1$ can be written as the sum of two squares, a power of 3, and a power of 5.

depends on sorry what this means
Statement uses: Nat, LT.lt, OfNat.ofNat, OeisA303656.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: OeisA303656.A