308734.lean`

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

7 declarations, 1 contain sorry, 0 declared as axioms

Four-square conjecture with powers of 2, 3, and 5

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

Zhi-Wei Sun has offered a $2,500 prize for the first proof.

References:

A308734
Z.-W. Sun, "Refining Lagrange's four-square theorem," J. Number Theory 175 (2017), 167-190. https://doi.org/10.1016/j.jnt.2016.11.008
Z.-W. Sun, "Restricted sums of four squares," Int. J. Number Theory 15 (2019), 1863-1893.
Z.-W. Sun, "Various Refinements of Lagrange's Four-Square Theorem," Westlake Number Theory Symposium, Nanjing University, China, 2020.
S. Banerjee, "On a conjecture of Sun about sums of restricted squares," J. Number Theory 256 (2024), 253-289.

Imports (1)

FormalConjecturesUtil

Imported by 1 file

FormalConjectures/Subsets/FC100OpenSet1.lean

def OeisA308734.A The predicate that n can be written as $(2^a \cdot 3^b)^2 + (2^c \cdot 5^d)^2 + x^2 + y^2$ for nonnegative integers $a, b, c, d, x, y$ line 45 source
Source documentation
The predicate that n can be written as $(2^a \cdot 3^b)^2 + (2^c \cdot 5^d)^2 + x^2 + y^2$ for nonnegative integers $a, b, c, d, x, y$.

standard axioms: propext what this means
Statement uses: Nat
Proof uses: Exists, Eq, HAdd.hAdd, HPow.hPow, HMul.hMul, OfNat.ofNat
9 instances, coercions and auxiliary constants left out.
```
def A (n : ℕ) : Prop :=
…
```
theorem OeisA308734.a_2 line 49 source
standard axioms: propext what this means
Statement uses: OeisA308734.A, OfNat.ofNat, Nat
Proof uses: Exists.intro, Exists, Eq, HAdd.hAdd, HPow.hPow, HMul.hMul, 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.isNat_mul, Mathlib.Meta.NormNum.natPow_zero, Mathlib.Meta.NormNum.one_natPow, Mathlib.Meta.NormNum.zero_natPow
17 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
theorem OeisA308734.a_3 line 53 source
standard axioms: propext what this means
Statement uses: OeisA308734.A, OfNat.ofNat, Nat
Proof uses: Exists.intro, Exists, Eq, HAdd.hAdd, HPow.hPow, HMul.hMul, 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.isNat_mul, Mathlib.Meta.NormNum.natPow_zero, Mathlib.Meta.NormNum.one_natPow, Mathlib.Meta.NormNum.zero_natPow
17 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
theorem OeisA308734.a_4 line 57 source
standard axioms: propext what this means
Statement uses: OeisA308734.A, OfNat.ofNat, Nat
Proof uses: Exists.intro, Exists, Eq, HAdd.hAdd, HPow.hPow, HMul.hMul, 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.isNat_mul, Mathlib.Meta.NormNum.natPow_zero, Mathlib.Meta.NormNum.one_natPow
17 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
theorem OeisA308734.a_5 line 61 source
standard axioms: propext what this means
Statement uses: OeisA308734.A, OfNat.ofNat, Nat
Proof uses: Exists.intro, Exists, Eq, HAdd.hAdd, HPow.hPow, HMul.hMul, 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.isNat_mul, Mathlib.Meta.NormNum.natPow_one, Mathlib.Meta.NormNum.natPow_zero, Mathlib.Meta.NormNum.IsNatPowT.run, Mathlib.Meta.NormNum.IsNatPowT.bit0, Mathlib.Meta.NormNum.one_natPow, Mathlib.Meta.NormNum.zero_natPow
17 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
theorem OeisA308734.a_6 line 65 source
standard axioms: propext what this means
Statement uses: OeisA308734.A, OfNat.ofNat, Nat
Proof uses: Exists.intro, Exists, Eq, HAdd.hAdd, HPow.hPow, HMul.hMul, 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.isNat_mul, Mathlib.Meta.NormNum.natPow_zero, Mathlib.Meta.NormNum.one_natPow, Mathlib.Meta.NormNum.IsNatPowT.run, Mathlib.Meta.NormNum.IsNatPowT.bit0, Mathlib.Meta.NormNum.zero_natPow
17 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
theorem OeisA308734.conjecture Zhi-Wei Sun's Four-Square Conjecture (A308734): Any integer sorry line 73 source
Source documentation
Zhi-Wei Sun's Four-Square Conjecture (A308734): Any integer $n > 1$ can be written as $(2^a \cdot 3^b)^2 + (2^c \cdot 5^d)^2 + x^2 + y^2$ for nonnegative integers $a, b, c, d, x, y$.

depends on sorry what this means
Statement uses: Nat, LT.lt, OfNat.ofNat, OeisA308734.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: OeisA308734.A