322.lean`

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L4DEL: Lean4 Domain Explanation Language

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Credit: The upstream authors and contributors retain credit for the project and mathematics. lean4-explainer only renders their pinned source and adds the separately disclosed explanation layer.

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

2 declarations, 1 contain sorry, 0 declared as axioms

Erdős Problem 322

References:

erdosproblems.com/322
[Er36] Erdős, Paul, On the Representation of an Integer as the Sum of k k-th Powers. J. London Math. Soc. (1936), 133-136.
[Er65b] Erdős, Paul, Some recent advances and current problems in number theory. Lectures on Modern Mathematics, Vol. III (1965), 196-244.
[Gu04] Guy, Richard K., Unsolved problems in number theory. (2004), xviii+437.
[Ma36] Mahler, Kurt, Note on Hypothesis K of Hardy and Littlewood. J. London Math. Soc. (1936), 136-138.

Imports (1)

FormalConjecturesUtil

def Erdos322.representationCount For k ≥ 3, the number of ordered representations of n as a sum of k many kth powers of positive integers line 41 source
Source documentation
For k ≥ 3, the number of ordered representations of n as a sum of k many kth powers of positive integers. The bases can be restricted to the interval from 1 to n, since x ≤ x ^ k for positive x.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat
Proof uses: Finset.card, Fin, HAdd.hAdd, OfNat.ofNat, Finset.filter, And, LT.lt, Eq, Finset.sum, Finset.univ, HPow.hPow
17 instances, coercions and auxiliary constants left out.
```
def representationCount (k n : ℕ) : ℕ :=
…
```
theorem Erdos322.erdos_322 Let $k\geq 3$ and $A\subset \mathbb{N}$ be the set of $k$th powers sorry line 49 source
Source documentation
Let $k\geq 3$ and $A\subset \mathbb{N}$ be the set of $k$th powers. What is the order of growth of $1_A^{(k)}(n)$, i.e. the number of representations of $n$ as the sum of $k$ many $k$th powers? Does there exist some $c>0$ and infinitely many $n$ such that $$1_A^{(k)}(n) >n^c?$$

depends on sorry what this means
Statement uses: Iff, True, Nat, LE.le, OfNat.ofNat, Exists, Real, And, GT.gt, Set.Infinite, Set.ofPred, LT.lt, HPow.hPow, Erdos322.representationCount
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 erdos_322 : answer(sorry) ↔
    ∀ k : ℕ, 3 ≤ k → ∃ c > (0 : ℝ),
      {n : ℕ | (n : ℝ) ^ c < representationCount k n}.Infinite := by
…
```
Mentions: c, Erdos322.representationCount