EulerSumOfPowers.lean`

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

3 declarations, 1 contain sorry, 0 declared as axioms

Euler's sum of powers conjecture

Reference: Wikipedia

Imports (1)

FormalConjecturesUtil

theorem EulerSumOfPowers.eulers_sum_of_powers_conjecture Euler's sum of powers conjecture states that for integers sorry line 39 source
Source documentation
Euler's sum of powers conjecture states that for integers $n > 1$ and $k > 1$, if the sum of $n$ positive integers each raised to the $k$-th power equals another integer raised to the $k$-th power, then $n ≥ k$.
The conjecture is known to be false for $k = 4$ and $k = 5$, but remains open for $k ≥ 6$.

depends on sorry what this means
Statement uses: Nat, LT.lt, OfNat.ofNat, Fin, GT.gt, Eq, Finset.sum, Finset.univ, HPow.hPow, LE.le
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
18 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem eulers_sum_of_powers_conjecture (n k b : ℕ) (hn : 1 < n) (hk : 5 < k) (a : Fin n → ℕ)
    (ha : ∀ i, a i > 0) (hsum : ∑ i, (a i) ^ k = b ^ k) : k ≤ n := by
…
```
theorem EulerSumOfPowers.eulers_sum_of_powers_conjecture.false_for_k4 Euler's sum of powers conjecture is false for $k=4$ (counterexample exists) line 45 source
Source documentation
Euler's sum of powers conjecture is false for $k=4$ (counterexample exists).

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Not, Nat, LT.lt, OfNat.ofNat, Fin, GT.gt, Eq, Finset.sum, Finset.univ, HPow.hPow, LE.le
Proof uses: Eq.mpr, Exists, And, id, Eq.trans, Mathlib.Tactic.Push.not_forall_eq, congrArg, funext, Exists.intro, True, congr, eq_true, Mathlib.Meta.NormNum.isNat_lt_true, Mathlib.Meta.NormNum.isNat_ofNat, Eq.refl, Bool, Bool.false, forall_congr, Mathlib.Meta.NormNum.IsNat.to_eq, Mathlib.Meta.NormNum.isNat_pow and 11 more
37 instances, coercions and auxiliary constants left out.

Tactic trace: 5 steps
Open to load the checked before-and-after goal states.
```
theorem eulers_sum_of_powers_conjecture.false_for_k4 : ¬ (∀ (n b : ℕ) (_ : 1 < n)
    (a: Fin n → ℕ) (_ : ∀ i, a i > 0) (_ : ∑ i, (a i) ^ 4 = b ^ 4), 4 ≤ n) := by
…
```
theorem EulerSumOfPowers.eulers_sum_of_powers_conjecture.false_for_k5 Euler's sum of powers conjecture is false for $k=5$ (counterexample exists) line 55 source
Source documentation
Euler's sum of powers conjecture is false for $k=5$ (counterexample exists).

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Not, Nat, LT.lt, OfNat.ofNat, Fin, GT.gt, Eq, Finset.sum, Finset.univ, HPow.hPow, LE.le
Proof uses: Eq.mpr, Exists, And, id, Eq.trans, Mathlib.Tactic.Push.not_forall_eq, congrArg, funext, Exists.intro, True, congr, eq_true, Mathlib.Meta.NormNum.isNat_lt_true, Mathlib.Meta.NormNum.isNat_ofNat, Eq.refl, Bool, Bool.false, forall_congr, Mathlib.Meta.NormNum.IsNat.to_eq, Mathlib.Meta.NormNum.isNat_pow and 12 more
37 instances, coercions and auxiliary constants left out.

Tactic trace: 5 steps
Open to load the checked before-and-after goal states.
```
theorem eulers_sum_of_powers_conjecture.false_for_k5 : ¬ (∀ (n b : ℕ) (_ : 1 < n)
    (a: Fin n → ℕ) (_ : ∀ i, a i > 0) (_ : ∑ i, (a i) ^ 5 = b ^ 5), 5 ≤ n) := by
…
```