270.lean`

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

5 declarations, 4 contain sorry, 0 declared as axioms

Erdős Problem 270

References:

erdosproblems.com/270
[ErGr80] Erdős, P. and Graham, R., Old and new problems and results in combinatorial number theory. Monographies de L'Enseignement Mathematique (1980).
[Ha75] Hansen, E. R., A Table of Series and Products. Prentice-Hall (1975), 87.
[CrKo25] T. Crmarić and V. Kovač, On the irrationality of certain super-polynomially decaying series. arXiv:2504.18712 (2025).
[Ko26] V. Kovač, solution of the linear case (2026).

Imports (1)

FormalConjecturesUtil

def Erdos270.series The series $\sum_{n\geq 1} \frac{1}{(n+1)\cdots (n+f(n))}$, indexed from $0$ line 41 source
Source documentation
The series $\sum_{n\geq 1} \frac{1}{(n+1)\cdots (n+f(n))}$, indexed from $0$.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Real
Proof uses: tsum, Inv.inv, Finset.prod, Finset.Icc, HAdd.hAdd, OfNat.ofNat, SummationFilter.unconditional
13 instances, coercions and auxiliary constants left out.
```
noncomputable def series (f : ℕ → ℕ) : ℝ :=
…
```
theorem Erdos270.erdos_270 sorry line 63 source
depends on sorry what this means
Statement uses: Iff, False, Nat, Filter.Tendsto, Filter.atTop, Irrational, Erdos270.series
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous, OfNat.ofNat
3 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
theorem Erdos270.erdos_270.variants.every_value sorry line 74 source
depends on sorry what this means
Statement uses: Real, LT.lt, OfNat.ofNat, Exists, Nat, And, Filter.Tendsto, Filter.atTop, HasSum, Inv.inv, Finset.prod, Finset.Icc, HAdd.hAdd, SummationFilter.unconditional
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
32 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
theorem Erdos270.erdos_270.variants.monotone Documented declaration: monotone sorry line 85 source
Source documentation
It is still possible that this sum is always irrational if $f$ is assumed to be non-decreasing; Crmarić and Kovač [CrKo25] show that the set of the possible values of such a sum has Lebesgue measure zero.

depends on sorry what this means
Statement uses: Iff, True, Nat, Monotone, Filter.Tendsto, Filter.atTop, Irrational, Erdos270.series
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous, OfNat.ofNat
3 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_270.variants.monotone : answer(sorry) ↔
    ∀ f : ℕ → ℕ, Monotone f → Tendsto f atTop atTop → Irrational (series f) := by
…
```
Mentions: Erdos270.series
theorem Erdos270.erdos_270.variants.linear The case $f(n)=n$ has a positive answer, as observed by Crmarić and Kovač [Ko26]: $\sum_{n\geq 1} n!/(2n)!$ is irrational sorry line 92 source
Source documentation
The case $f(n)=n$ has a positive answer, as observed by Crmarić and Kovač [Ko26]: $\sum_{n\geq 1} n!/(2n)!$ is irrational.

depends on sorry what this means
Statement uses: Iff, True, Irrational, Erdos270.series, id, Nat
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous, OfNat.ofNat
1 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_270.variants.linear : answer(True) ↔ Irrational (series id) := by
…
```
Mentions: Erdos270.series