LanderParkinAndSelfridgeConjecture.lean`

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

2 declarations, 2 contain sorry, 0 declared as axioms

Lander, Parkin, and Selfridge Conjecture

Reference: https://en.wikipedia.org/wiki/Lander,_Parkin,_and_Selfridge_conjecture

Imports (1)

FormalConjecturesUtil

theorem LanderParkinSelfridge.lander_parkin_selfridge Documented declaration: lander_parkin_selfridge sorry line 40 source
Source documentation
The Lander–Parkin–Selfridge conjecture: if the sum of $n$ positive integer $k$-th powers equals the sum of $m$ positive integer $k$-th powers, with all values on the left distinct from all values on the right, then $n + m \geq k$.
Formally, for positive integers $k, n, m \in \mathbb{N}$ and sequences $x : \{0, \ldots, n-1\} \to \mathbb{N}$ and $y : \{0, \ldots, m-1\} \to \mathbb{N}$ with $x_i > 0$, $y_j > 0$, and $x_i \neq y_j$ for all $i, j$, if $$\sum_{i=0}^{n-1} x_i^k = \sum_{j=0}^{m-1} y_j^k,$$ then $k \leq n + m$.

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

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem lander_parkin_selfridge :
    ∀ (k n m : ℕ) (x : Fin n → ℕ) (y : Fin m → ℕ),
      0 < n → 0 < m →
      (∀ i, 0 < x i) → (∀ j, 0 < y j) →
      (∀ i j, x i ≠ y j) →
      ∑ i, x i ^ k = ∑ j, y j ^ k →
      k ≤ n + m := by
…
```
theorem LanderParkinSelfridge.lander_parkin_selfridge.variants.five_three Documented declaration: five_three sorry line 57 source
Source documentation
Special case of the Lander–Parkin–Selfridge conjecture: there is no solution in positive integers to $$x_1^5 + x_2^5 + x_3^5 = y^5.$$ That is, for all $x_1, x_2, x_3, y \in \mathbb{N}$ with $x_1, x_2, x_3, y > 0$, $$x_1^5 + x_2^5 + x_3^5 \neq y^5.$$ This corresponds to the case $k = 5$, $n = 3$, $m = 1$ of the general conjecture, where $n + m = 4 < 5 = k$ would be required to yield a counterexample.

depends on sorry what this means
Statement uses: Nat, LT.lt, OfNat.ofNat, Ne, HAdd.hAdd, HPow.hPow
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
16 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem lander_parkin_selfridge.variants.five_three :
    ∀ x₁ x₂ x₃ y : ℕ,
      0 < x₁ → 0 < x₂ → 0 < x₃ → 0 < y →
      x₁ ^ 5 + x₂ ^ 5 + x₃ ^ 5 ≠ y ^ 5 := by
…
```