1050.lean`

Explanation legend Lean4 Domain Explanation Language (L4DEL)

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L4DEL: Lean4 Domain Explanation Language

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

4 declarations, 4 contain sorry, 0 declared as axioms

Erdős Problem 1050

References:

erdosproblems.com/1050
[Bo91] Borwein, Peter B., On the irrationality of {$\sum(1/(q^n+r))$}. J. Number Theory (1991), 253--259.
[Bo92] Borwein, Peter B., On the irrationality of certain series. Math. Proc. Camb. Phil. Soc. (1992), 141--146.
[Er48] Erdős, P., On arithmetical properties of Lambert series. J. Indian Math. Soc. (N.S.) (1948), 63-66.
[Er88c] Erdős, P., On the irrationality of certain series: problems and results. New advances in transcendence theory (Durham, 1986) (1988), 102-109.

Is $\sum_{n=1}^\infty \frac{1}{2^n - 3}$ irrational?

The answer is yes, proved by P. B. Borwein [Bo91] (with a cleaner self-contained proof in [Bo92]), specialized to $q = 2$, $r = -3$.

A formal Lean proof is given in an external repository, gotrevor/lean-gallery, formalized by Trevor Morris with Claude Code and Harmonic's Aristotle.

Imports (1)

FormalConjecturesUtil

theorem Erdos1050.erdos_1050 sorry line 51 source
depends on sorry what this means
Statement uses: Irrational, tsum, Real, Nat, HDiv.hDiv, OfNat.ofNat, HSub.hSub, HPow.hPow, HAdd.hAdd, SummationFilter.unconditional
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
44 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
theorem Erdos1050.erdos_1050.variants.two_pow_sub_one sorry line 61 source
depends on sorry what this means
Statement uses: Irrational, tsum, Real, Nat, HDiv.hDiv, OfNat.ofNat, HSub.hSub, HPow.hPow, HAdd.hAdd, SummationFilter.unconditional
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
44 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
theorem Erdos1050.erdos_1050.variants.borwein sorry line 71 source
depends on sorry what this means
Statement uses: Int, LE.le, OfNat.ofNat, Rat, Ne, Nat, Neg.neg, HPow.hPow, Irrational, tsum, Real, HDiv.hDiv, HAdd.hAdd, SummationFilter.unconditional
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
56 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
theorem Erdos1050.erdos_1050.variants.transcendental Erdős Problem 1050, variant [Er88c] (open) sorry line 84 source
Source documentation
Erdős Problem 1050, variant [Er88c] (open). Erdős conjectured that $\sum_{n=1}^\infty \frac{1}{2^n + t}$ is transcendental for every integer $t \ne 0$ — strictly stronger than the irrationality established by Borwein [Bo91]. This remains open.
Two exclusions make the series well-posed and the claim non-vacuous, exactly as in Borwein's theorem: $t \ne -2^n$ for all $n \ge 1$ (so no denominator vanishes), and $t \ne 0$ (at $t = 0$ the series is the rational $\sum_{n \ge 1} 2^{-n} = 1$, hence not transcendental).

depends on sorry what this means
Statement uses: Iff, True, Int, Ne, OfNat.ofNat, Nat, LE.le, Neg.neg, HPow.hPow, Transcendental, Rat, Real, tsum, HDiv.hDiv, HAdd.hAdd, SummationFilter.unconditional
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
76 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_1050.variants.transcendental :
    answer(sorry) ↔
      ∀ t : ℤ, t ≠ 0 → (∀ n : ℕ, 1 ≤ n → t ≠ -(2 : ℤ) ^ n) →
        Transcendental ℚ (∑' n : ℕ, (1 : ℝ) / ((2 : ℝ) ^ (n + 1) + (t : ℝ))) := by
…
```