678.lean`

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

7 declarations, 3 contain sorry, 0 declared as axioms

Erdős Problem 678

References:

erdosproblems.com/678
[Ca24] S. Cambie, Resolution of an Erdős' problem on least common multiples. arXiv:2410.09138 (2024).
[Er79] Erdős, Paul, Some unconventional problems in number theory. Math. Mag. (1979), 67-70.
[Er92e] Erdős, Pál, Some Unsolved problems in Geometry, Number Theory and Combinatorics. Eureka (1992), 44-48.

Imports (1)

FormalConjecturesUtil

Imported by 1 file

FormalConjectures/Subsets/FC100SolvedSet1.lean

lemma Erdos678.lcmInterval_lt_example1 The referee of [Er79] found the example $M(96, 7) > M(104, 8)$, showing that there are cases where $M(n, k) > M(m, k + 1)$ with $m \geq n + k$ line 43 source
Source documentation
The referee of [Er79] found the example $M(96, 7) > M(104, 8)$, showing that there are cases where $M(n, k) > M(m, k + 1)$ with $m \geq n + k$. [Er79] Erdős, Paul, Some unconventional problems in number theory. Math. Mag. (1979), 67-70.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: LT.lt, Nat, Finset.lcmInterval, OfNat.ofNat
Proof uses: of_decide_eq_true, id, Eq, Bool, Decidable.decide, Bool.true, Eq.refl
17 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
lemma lcmInterval_lt_example1 : lcmInterval 104 8 < lcmInterval 96 7 := by decide
…
```
lemma Erdos678.lcmInterval_lt_example2 The referee of [Er79] found the example $M(132, 7) > M(139, 8)$, showing that there are cases where $M(n, k) > M(m, k + 1)$ with $m \geq n + k$ line 51 source
Source documentation
The referee of [Er79] found the example $M(132, 7) > M(139, 8)$, showing that there are cases where $M(n, k) > M(m, k + 1)$ with $m \geq n + k$. [Er79] Erdős, Paul, Some unconventional problems in number theory. Math. Mag. (1979), 67-70.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: LT.lt, Nat, Finset.lcmInterval, OfNat.ofNat
Proof uses: of_decide_eq_true, id, Eq, Bool, Decidable.decide, Bool.true, Eq.refl
17 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
lemma lcmInterval_lt_example2 : lcmInterval 139 8 < lcmInterval 132 7 := by decide
…
```
lemma Erdos678.lcmInterval_lt_example3 Cambie [Ca24] found the example $M(52, 7) > M(62, 8)$ line 58 source
Source documentation
Cambie [Ca24] found the example $M(52, 7) > M(62, 8)$. [Ca24] S. Cambie, Resolution of an Erdős' problem on least common multiples. arXiv:2410.09138 (2024).

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: LT.lt, Nat, Finset.lcmInterval, OfNat.ofNat
Proof uses: of_decide_eq_true, id, Eq, Bool, Decidable.decide, Bool.true, Eq.refl
17 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
lemma lcmInterval_lt_example3 : lcmInterval 62 8 < lcmInterval 52 7 := by decide
…
```
lemma Erdos678.lcmInterval_lt_example4 Cambie [Ca24] found the example $M(36, 8) > M(48, 9)$ line 65 source
Source documentation
Cambie [Ca24] found the example $M(36, 8) > M(48, 9)$. [Ca24] S. Cambie, Resolution of an Erdős' problem on least common multiples. arXiv:2410.09138 (2024).

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: LT.lt, Nat, Finset.lcmInterval, OfNat.ofNat
Proof uses: of_decide_eq_true, id, Eq, Bool, Decidable.decide, Bool.true, Eq.refl
17 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
lemma lcmInterval_lt_example4 : lcmInterval 47 9 < lcmInterval 36 8 := by decide
…
```
theorem Erdos678.erdos_678 Write $M(n, k)$ be the least common multiple of $\{n+1, \dotsc, n+k\}$ sorry line 85 source
Source documentation
Write $M(n, k)$ be the least common multiple of $\{n+1, \dotsc, n+k\}$. Let $k$ be sufficiently large. Are there infinitely many $m, n$ with $m \geq n + k$ such that $$ M(n, k) > M(m, k + 1) $$? The answer is yes, as proved in a strong form by Cambie [Ca24]. [Ca24] S. Cambie, Resolution of an Erdős' problem on least common multiples. arXiv:2410.09138 (2024).
This was formalized in Lean by Alexeev using Aristotle, on top of the PNT+ project.
For a fixed $k$ there are only finitely many such pairs, so "infinitely many" is read here as ranging over $k$ as well: for every sufficiently large $k$ at least one pair $(m, n)$ occurs. See erdos_678.variants.infinitely_many_triples for the reading in which the infinitude is stated directly, and erdos_678.variants.not_infinitely_many_pairs for why it cannot be asked of a single $k$.

depends on sorry what this means
Statement uses: Iff, True, Filter.Eventually, Nat, Set.Nonempty, Prod, Set.ofPred, Erdos678.erdos_678.match_1, And, LE.le, HAdd.hAdd, LT.lt, Finset.lcmInterval, OfNat.ofNat, Filter.atTop
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 erdos_678 : answer(True) ↔
    ∀ᶠ k in atTop, {(m, n) | n + k ≤ m ∧ lcmInterval m (k + 1) < lcmInterval n k}.Nonempty := by
…
```
theorem Erdos678.erdos_678.variants.infinitely_many_triples Documented declaration: infinitely_many_triples sorry line 94 source
Source documentation
The pairs $(m, n)$ with $m \geq n + k$ and $M(n, k) > M(m, k + 1)$ are infinite in number once $k$ is allowed to vary, which is the sense in which Cambie's result answers the question.

depends on sorry what this means
Statement uses: Set.Infinite, Prod, Nat, Set.ofPred, Erdos678.erdos_678.variants.infinitely_many_triples.match_1, And, LE.le, OfNat.ofNat, HAdd.hAdd, LT.lt, Finset.lcmInterval
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 erdos_678.variants.infinitely_many_triples :
    {(k, m, n) | 3 ≤ k ∧ n + k ≤ m ∧ lcmInterval m (k + 1) < lcmInterval n k}.Infinite := by
…
```
theorem Erdos678.erdos_678.variants.not_infinitely_many_pairs For a fixed sufficiently large $k$ only finitely many pairs occur: $M(m, k + 1) \geq m + 1$ bounds $m$ by $M(n, k)$, and for large $n$ the inequality reverses sorry line 104 source
Source documentation
For a fixed sufficiently large $k$ only finitely many pairs occur: $M(m, k + 1) \geq m + 1$ bounds $m$ by $M(n, k)$, and for large $n$ the inequality reverses. So the question cannot be read as asking for infinitely many pairs at a single $k$.

depends on sorry what this means
Statement uses: Not, Filter.Eventually, Nat, Set.Infinite, Prod, Set.ofPred, Erdos678.erdos_678.match_1, And, LE.le, HAdd.hAdd, LT.lt, Finset.lcmInterval, OfNat.ofNat, Filter.atTop
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 erdos_678.variants.not_infinitely_many_pairs :
    ¬ ∀ᶠ k in atTop, {(m, n) | n + k ≤ m ∧ lcmInterval m (k + 1) < lcmInterval n k}.Infinite := by
…
```