CongruentNumber.lean`

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14 declarations, 5 contain sorry, 0 declared as axioms

Congruent Number

A natural number $n$ is called a congruent number if there exists a right triangle with rational sides $a$, $b$, and hypotenuse $c$ such that the area of the triangle is $\frac{1}{2}ab = n$.

References:

Imports (1)

FormalConjecturesUtil

Imported by 1 file

FormalConjectures/Subsets/FC100SolvedSet1.lean

def CongruentNumber.congruentNumber line 36 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat
Proof uses: Exists, Rat, And, Eq, HAdd.hAdd, HPow.hPow, OfNat.ofNat, HMul.hMul, Inv.inv
13 instances, coercions and auxiliary constants left out.
theorem CongruentNumber.not_congruentNumber_1 1 is not a congruent number, as proved by Fermat via infinite descent sorry line 41 source
Source documentation
1 is not a congruent number, as proved by Fermat via infinite descent.

depends on sorry what this means
Statement uses: Not, CongruentNumber.congruentNumber, OfNat.ofNat, Nat
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
2 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem not_congruentNumber_1 : ¬ congruentNumber 1 := by
…
```
Mentions: CongruentNumber.congruentNumber
theorem CongruentNumber.congruentNumber_5 The rational right triangle with side lengths $\frac{3}{2}$, $\frac{20}{3}$, and $\frac{41}{6}$ witnesses that $5$ is a congruent number line 49 source
Source documentation
The rational right triangle with side lengths $\frac{3}{2}$, $\frac{20}{3}$, and $\frac{41}{6}$ witnesses that $5$ is a congruent number.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: CongruentNumber.congruentNumber, OfNat.ofNat, Nat
Proof uses: Exists.intro, Rat, Exists, And, Eq, HAdd.hAdd, HPow.hPow, HMul.hMul, Inv.inv, HDiv.hDiv, of_eq_true, Eq.trans, True, congr, congrArg, eq_true, Mathlib.Meta.NormNum.isNNRat_eq_true, Mathlib.Meta.NormNum.isNNRat_add, Eq.refl, Mathlib.Meta.NormNum.isNNRat_pow and 13 more
35 instances, coercions and auxiliary constants left out.

Tactic trace: 2 steps
Open to load the checked before-and-after goal states.
```
theorem congruentNumber_5 : congruentNumber 5 := by
…
```
Mentions: CongruentNumber.congruentNumber
theorem CongruentNumber.congruentNumber_6 The $3$-$4$-$5$ right triangle witnesses that $6$ is a congruent number line 57 source
Source documentation
The $3$-$4$-$5$ right triangle witnesses that $6$ is a congruent number.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: CongruentNumber.congruentNumber, OfNat.ofNat, Nat
Proof uses: Exists.intro, Rat, Exists, And, Eq, HAdd.hAdd, HPow.hPow, HMul.hMul, Inv.inv, of_eq_true, Eq.trans, True, congr, congrArg, eq_true, Mathlib.Meta.NormNum.isNat_eq_true, Mathlib.Meta.NormNum.isNat_add, Eq.refl, Mathlib.Meta.NormNum.isNat_pow, Mathlib.Meta.NormNum.isNat_ofNat and 10 more
30 instances, coercions and auxiliary constants left out.

Tactic trace: 2 steps
Open to load the checked before-and-after goal states.
```
theorem congruentNumber_6 : congruentNumber 6 := by
…
```
Mentions: CongruentNumber.congruentNumber
theorem CongruentNumber.congruentNumber_7 The rational right triangle with side lengths $\frac{35}{12}$, $\frac{24}{5}$, and $\frac{337}{60}$ witnesses that $7$ is a congruent number line 66 source
Source documentation
The rational right triangle with side lengths $\frac{35}{12}$, $\frac{24}{5}$, and $\frac{337}{60}$ witnesses that $7$ is a congruent number.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: CongruentNumber.congruentNumber, OfNat.ofNat, Nat
Proof uses: Exists.intro, Rat, Exists, And, Eq, HAdd.hAdd, HPow.hPow, HMul.hMul, Inv.inv, HDiv.hDiv, of_eq_true, Eq.trans, True, congr, congrArg, eq_true, Mathlib.Meta.NormNum.isNNRat_eq_true, Mathlib.Meta.NormNum.isNNRat_add, Eq.refl, Mathlib.Meta.NormNum.isNNRat_pow and 13 more
35 instances, coercions and auxiliary constants left out.

Tactic trace: 2 steps
Open to load the checked before-and-after goal states.
```
theorem congruentNumber_7 : congruentNumber 7 := by
…
```
Mentions: CongruentNumber.congruentNumber
theorem CongruentNumber.congruentNumber_157_zagier Zagier's rational right triangle witnesses that $157$ is a congruent number line 74 source
Source documentation
Zagier's rational right triangle witnesses that $157$ is a congruent number.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: CongruentNumber.congruentNumber, OfNat.ofNat, Nat
Proof uses: Exists.intro, Rat, Exists, And, Eq, HAdd.hAdd, HPow.hPow, HMul.hMul, Inv.inv, HDiv.hDiv, of_eq_true, Eq.trans, True, congr, congrArg, eq_true, Mathlib.Meta.NormNum.isNNRat_eq_true, Mathlib.Meta.NormNum.isNNRat_add, Eq.refl, Mathlib.Meta.NormNum.isNNRat_pow and 13 more
35 instances, coercions and auxiliary constants left out.

Tactic trace: 2 steps
Open to load the checked before-and-after goal states.
```
theorem congruentNumber_157_zagier : congruentNumber 157 := by
…
```
Mentions: CongruentNumber.congruentNumber
def CongruentNumber.A line 95 source
standard axioms: propext what this means
Statement uses: Nat, Set, Prod, Int
Proof uses: Set.ofPred, CongruentNumber.A.match_1, Eq, HAdd.hAdd, HMul.hMul, OfNat.ofNat, HPow.hPow
12 instances, coercions and auxiliary constants left out.
def CongruentNumber.B line 96 source
standard axioms: propext what this means
Statement uses: Nat, Set, Prod, Int
Proof uses: Set.ofPred, CongruentNumber.A.match_1, Eq, HAdd.hAdd, HMul.hMul, OfNat.ofNat, HPow.hPow
12 instances, coercions and auxiliary constants left out.
def CongruentNumber.C line 97 source
standard axioms: propext what this means
Statement uses: Nat, Set, Prod, Int
Proof uses: Set.ofPred, CongruentNumber.A.match_1, Eq, HAdd.hAdd, HMul.hMul, OfNat.ofNat, HPow.hPow
12 instances, coercions and auxiliary constants left out.
def CongruentNumber.D line 98 source
standard axioms: propext what this means
Statement uses: Nat, Set, Prod, Int
Proof uses: Set.ofPred, CongruentNumber.A.match_1, Eq, HAdd.hAdd, HMul.hMul, OfNat.ofNat, HPow.hPow
12 instances, coercions and auxiliary constants left out.
theorem CongruentNumber.Tunnell_odd Tunnell's theorem (necessary condition) for odd squarefree congruent numbers sorry line 104 source
Source documentation
Tunnell's theorem (necessary condition) for odd squarefree congruent numbers.

depends on sorry what this means
Statement uses: Nat, Squarefree, Odd, CongruentNumber.congruentNumber, Eq, HMul.hMul, OfNat.ofNat, Set.ncard, Prod, Int, CongruentNumber.A, CongruentNumber.B
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
10 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem Tunnell_odd (n : ℕ) (hsqf : Squarefree n) (hodd : Odd n) :
    congruentNumber n → 2 * (A n).ncard = (B n).ncard := by
…
```
Mentions: CongruentNumber.congruentNumber, CongruentNumber.A, CongruentNumber.B
theorem CongruentNumber.Tunnell_even Tunnell's theorem (necessary condition) for even squarefree congruent numbers sorry line 110 source
Source documentation
Tunnell's theorem (necessary condition) for even squarefree congruent numbers.

depends on sorry what this means
Statement uses: Nat, Squarefree, Even, CongruentNumber.congruentNumber, Eq, HMul.hMul, OfNat.ofNat, Set.ncard, Prod, Int, CongruentNumber.C, CongruentNumber.D
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
10 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem Tunnell_even (n : ℕ) (hsqf : Squarefree n) (heven : Even n) :
    congruentNumber n → 2 * (C n).ncard = (D n).ncard := by
…
```
Mentions: CongruentNumber.congruentNumber, CongruentNumber.C, CongruentNumber.D
theorem CongruentNumber.Tunnell_odd_converse Tunnell's theorem (sufficient condition assuming BSD) for odd squarefree congruent numbers sorry line 118 source
Source documentation
Tunnell's theorem (sufficient condition assuming BSD) for odd squarefree congruent numbers.

depends on sorry what this means
Statement uses: Nat, Squarefree, Odd, Eq, HMul.hMul, OfNat.ofNat, Set.ncard, Prod, Int, CongruentNumber.A, CongruentNumber.B, CongruentNumber.congruentNumber
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
10 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem Tunnell_odd_converse (n : ℕ) (hsqf : Squarefree n) (hodd : Odd n) :
    2 * (A n).ncard = (B n).ncard → congruentNumber n := by
…
```
Mentions: CongruentNumber.A, CongruentNumber.B, CongruentNumber.congruentNumber
theorem CongruentNumber.Tunnell_even_converse Tunnell's theorem (sufficient condition assuming BSD) for even squarefree congruent numbers sorry line 124 source
Source documentation
Tunnell's theorem (sufficient condition assuming BSD) for even squarefree congruent numbers.

depends on sorry what this means
Statement uses: Nat, Squarefree, Even, Eq, HMul.hMul, OfNat.ofNat, Set.ncard, Prod, Int, CongruentNumber.C, CongruentNumber.D, CongruentNumber.congruentNumber
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
10 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem Tunnell_even_converse (n : ℕ) (hsqf : Squarefree n) (heven : Even n) :
    2 * (C n).ncard = (D n).ncard → congruentNumber n := by
…
```
Mentions: CongruentNumber.C, CongruentNumber.D, CongruentNumber.congruentNumber