IdonealCompleteness.lean`

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5 declarations, 1 contain sorry, 0 declared as axioms

Idoneal numbers completeness conjecture

An integer $D>0$ is idoneal if every integer that can be expressed in exactly one way (up to order and signs) as $x^2 + D y^2$ with gcd(x, Dy)=1 is a prime power or twice a prime power.

The Idoneal Numbers Completeness Conjecture asserts that the following list of 65 numbers is complete: 1,2,3,4,5,6,7,8,9,10,12,13,15,16,18,21,22,24,25,28,30,33,37,40,42,45,48, 57,58,60,70,72,78,85,88,93,102,105,112,120,130,133,165,168,177,190,210,232, 240,253,273,280,312,330,345,357,385,408,462,520,760,840,1320,1365,1848. References:

Imports (1)

FormalConjecturesUtil

def Idoneal.IsIdoneal Equivalent definition: A positive integer $n$ is idoneal if and only if it cannot be written as $ab + bc + ac$ for distinct positive integers $a, b,$ and $c$ line 45 source
Source documentation
Equivalent definition: A positive integer $n$ is idoneal if and only if it cannot be written as $ab + bc + ac$ for distinct positive integers $a, b,$ and $c$.

no axioms what this means
Statement uses: Nat
Proof uses: And, LT.lt, OfNat.ofNat, Not, Exists, Eq, HAdd.hAdd, HMul.hMul
6 instances, coercions and auxiliary constants left out.
```
def IsIdoneal (n : ℕ) : Prop :=
…
```
def Idoneal.knownIdonealNumbers The 65 known idoneal numbers that are conjectured to be the only idoneal numbers line 53 source
Source documentation
The 65 known idoneal numbers that are conjectured to be the only idoneal numbers.

standard axioms: Quot.sound, propext what this means
Statement uses: Finset, Nat
Proof uses: Insert.insert, OfNat.ofNat, Singleton.singleton
4 instances, coercions and auxiliary constants left out.
```
def knownIdonealNumbers : Finset ℕ :=
…
```
theorem Idoneal.exists_triple_iff_bounded Reduces the unbounded search for a representation n = a*b + b*c + a*c (with 0 < a < b < c) to a bounded, decidable double search over a, b ∈ range (n+1) line 69 source
Source documentation
Reduces the unbounded search for a representation n = a*b + b*c + a*c (with 0 < a < b < c) to a bounded, decidable double search over a, b ∈ range (n+1).
The third variable is not searched: for a fixed pair a, b the equation n = a*b + c*(a+b) pins down c = (n - a*b) / (a+b), so the witness c is recovered by exact division. The forward direction uses a, b ≤ n (each pairwise product is at most n) to land the pair in range (n+1).
```
private theorem exists_triple_iff_bounded (n : ℕ) :
    (∃ a b c : ℕ, 0 < a ∧ a < b ∧ b < c ∧ n = a * b + b * c + a * c) ↔
      (∃ a ∈ Finset.range (n + 1), ∃ b ∈ Finset.range (n + 1),
        0 < a ∧ a < b ∧ b < (n - a * b) / (a + b) ∧
          n = a * b + b * ((n - a * b) / (a + b)) + a * ((n - a * b) / (a + b))) := by
…
```
Mentions: c
theorem Idoneal.knownIdonealNumbers_are_idoneal All 65 known idoneal numbers are indeed idoneal line 91 source
Source documentation
All 65 known idoneal numbers are indeed idoneal.

extra axioms: _private.0.Idoneal.knownIdonealNumbers_are_idoneal._native.native_decide.ax_1, _private.0.Idoneal.knownIdonealNumbers_are_idoneal._native.native_decide.ax_10, _private.0.Idoneal.knownIdonealNumbers_are_idoneal._native.native_decide.ax_11, _private.0.Idoneal.knownIdonealNumbers_are_idoneal._native.native_decide.ax_12, _private.0.Idoneal.knownIdonealNumbers_are_idoneal._native.native_decide.ax_13, _private.0.Idoneal.knownIdonealNumbers_are_idoneal._native.native_decide.ax_14, _private.0.Idoneal.knownIdonealNumbers_are_idoneal._native.native_decide.ax_15, _private.0.Idoneal.knownIdonealNumbers_are_idoneal._native.native_decide.ax_16, _private.0.Idoneal.knownIdonealNumbers_are_idoneal._native.native_decide.ax_17, _private.0.Idoneal.knownIdonealNumbers_are_idoneal._native.native_decide.ax_18, _private.0.Idoneal.knownIdonealNumbers_are_idoneal._native.native_decide.ax_19, _private.0.Idoneal.knownIdonealNumbers_are_idoneal._native.native_decide.ax_2, _private.0.Idoneal.knownIdonealNumbers_are_idoneal._native.native_decide.ax_20, _private.0.Idoneal.knownIdonealNumbers_are_idoneal._native.native_decide.ax_21, _private.0.Idoneal.knownIdonealNumbers_are_idoneal._native.native_decide.ax_22, _private.0.Idoneal.knownIdonealNumbers_are_idoneal._native.native_decide.ax_23, _private.0.Idoneal.knownIdonealNumbers_are_idoneal._native.native_decide.ax_24, _private.0.Idoneal.knownIdonealNumbers_are_idoneal._native.native_decide.ax_25, _private.0.Idoneal.knownIdonealNumbers_are_idoneal._native.native_decide.ax_26, _private.0.Idoneal.knownIdonealNumbers_are_idoneal._native.native_decide.ax_27, _private.0.Idoneal.knownIdonealNumbers_are_idoneal._native.native_decide.ax_28, _private.0.Idoneal.knownIdonealNumbers_are_idoneal._native.native_decide.ax_29, _private.0.Idoneal.knownIdonealNumbers_are_idoneal._native.native_decide.ax_3, _private.0.Idoneal.knownIdonealNumbers_are_idoneal._native.native_decide.ax_30, _private.0.Idoneal.knownIdonealNumbers_are_idoneal._native.native_decide.ax_31, _private.0.Idoneal.knownIdonealNumbers_are_idoneal._native.native_decide.ax_32, _private.0.Idoneal.knownIdonealNumbers_are_idoneal._native.native_decide.ax_33, _private.0.Idoneal.knownIdonealNumbers_are_idoneal._native.native_decide.ax_34, _private.0.Idoneal.knownIdonealNumbers_are_idoneal._native.native_decide.ax_35, _private.0.Idoneal.knownIdonealNumbers_are_idoneal._native.native_decide.ax_36, _private.0.Idoneal.knownIdonealNumbers_are_idoneal._native.native_decide.ax_37, _private.0.Idoneal.knownIdonealNumbers_are_idoneal._native.native_decide.ax_38, _private.0.Idoneal.knownIdonealNumbers_are_idoneal._native.native_decide.ax_39, _private.0.Idoneal.knownIdonealNumbers_are_idoneal._native.native_decide.ax_4, _private.0.Idoneal.knownIdonealNumbers_are_idoneal._native.native_decide.ax_40, _private.0.Idoneal.knownIdonealNumbers_are_idoneal._native.native_decide.ax_41, _private.0.Idoneal.knownIdonealNumbers_are_idoneal._native.native_decide.ax_42, _private.0.Idoneal.knownIdonealNumbers_are_idoneal._native.native_decide.ax_43, _private.0.Idoneal.knownIdonealNumbers_are_idoneal._native.native_decide.ax_44, _private.0.Idoneal.knownIdonealNumbers_are_idoneal._native.native_decide.ax_45, _private.0.Idoneal.knownIdonealNumbers_are_idoneal._native.native_decide.ax_46, _private.0.Idoneal.knownIdonealNumbers_are_idoneal._native.native_decide.ax_47, _private.0.Idoneal.knownIdonealNumbers_are_idoneal._native.native_decide.ax_48, _private.0.Idoneal.knownIdonealNumbers_are_idoneal._native.native_decide.ax_49, _private.0.Idoneal.knownIdonealNumbers_are_idoneal._native.native_decide.ax_5, _private.0.Idoneal.knownIdonealNumbers_are_idoneal._native.native_decide.ax_50, _private.0.Idoneal.knownIdonealNumbers_are_idoneal._native.native_decide.ax_51, _private.0.Idoneal.knownIdonealNumbers_are_idoneal._native.native_decide.ax_52, _private.0.Idoneal.knownIdonealNumbers_are_idoneal._native.native_decide.ax_53, _private.0.Idoneal.knownIdonealNumbers_are_idoneal._native.native_decide.ax_54, _private.0.Idoneal.knownIdonealNumbers_are_idoneal._native.native_decide.ax_55, _private.0.Idoneal.knownIdonealNumbers_are_idoneal._native.native_decide.ax_56, _private.0.Idoneal.knownIdonealNumbers_are_idoneal._native.native_decide.ax_57, _private.0.Idoneal.knownIdonealNumbers_are_idoneal._native.native_decide.ax_58, _private.0.Idoneal.knownIdonealNumbers_are_idoneal._native.native_decide.ax_59, _private.0.Idoneal.knownIdonealNumbers_are_idoneal._native.native_decide.ax_6, _private.0.Idoneal.knownIdonealNumbers_are_idoneal._native.native_decide.ax_60, _private.0.Idoneal.knownIdonealNumbers_are_idoneal._native.native_decide.ax_61, _private.0.Idoneal.knownIdonealNumbers_are_idoneal._native.native_decide.ax_62, _private.0.Idoneal.knownIdonealNumbers_are_idoneal._native.native_decide.ax_63, _private.0.Idoneal.knownIdonealNumbers_are_idoneal._native.native_decide.ax_64, _private.0.Idoneal.knownIdonealNumbers_are_idoneal._native.native_decide.ax_65, _private.0.Idoneal.knownIdonealNumbers_are_idoneal._native.native_decide.ax_7, _private.0.Idoneal.knownIdonealNumbers_are_idoneal._native.native_decide.ax_8, _private.0.Idoneal.knownIdonealNumbers_are_idoneal._native.native_decide.ax_9 what this means
Statement uses: Nat, Membership.mem, Finset, Idoneal.knownIdonealNumbers, Idoneal.IsIdoneal
Proof uses: List, List.Mem, Eq, List.insert, OfNat.ofNat, List.cons, List.nil, HEq, List.Mem.head, And.intro, LT.lt, Not, Exists, And, HAdd.hAdd, HMul.hMul, Mathlib.Meta.NormNum.isNat_lt_true, Mathlib.Meta.NormNum.isNat_ofNat, Eq.refl, Bool and 15 more
97 instances, coercions and auxiliary constants left out.

Tactic trace: 6 steps
Open to load the checked before-and-after goal states.
```
theorem knownIdonealNumbers_are_idoneal : ∀ n ∈ knownIdonealNumbers, IsIdoneal n := by
…
```
Mentions: Idoneal.knownIdonealNumbers, Idoneal.IsIdoneal
theorem Idoneal.idoneal_numbers_completeness Idoneal numbers completeness conjecture sorry line 102 source
Source documentation
Idoneal numbers completeness conjecture.

depends on sorry what this means
Statement uses: Iff, True, Nat, Idoneal.IsIdoneal, Membership.mem, Finset, Idoneal.knownIdonealNumbers
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous, OfNat.ofNat
5 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem idoneal_numbers_completeness :
    answer(sorry) ↔
      ∀ n : ℕ, IsIdoneal n → n ∈ knownIdonealNumbers := by
…
```
Mentions: Idoneal.IsIdoneal, Idoneal.knownIdonealNumbers