Is the totient series irrational?
Sum φ(n)/2ⁿ over all n, where φ counts integers coprime to n. Is the result irrational?
▸Motivation
Erdős problem 249, catalogued at erdosproblems.com/249.
The series converges quickly, so the question is entirely about the arithmetic of the totient function rather than about convergence. Irrationality proofs for series like this usually need either a good rational approximation argument or strong control of the digits; neither is available here.
Erdős problem 249. Lean statement adapted from google-deepmind/formal-conjectures (Apache 2.0).
▸Lean API
import Mathlib.NumberTheory.Real.Irrational
import Mathlib.Data.Nat.Totient
import Mathlib.Topology.Algebra.InfiniteSum.Defs
namespace Conjectura.NT007
/-- Is `∑ φ(n) / 2ⁿ` irrational, where `φ` is Euler's totient function? -/
def goal : Prop := Irrational (∑' n : ℕ, (φ n : ℝ) / (2 ^ n))
end Conjectura.NT007▸Definition1
- IrrationalIrrational
def
▸Related work2
- Erdős problem 249Thomas Bloom (catalogue)
The catalogue entry, with references and current status.
- formal-conjecturesThe Formal Conjectures Authors (Google DeepMind) · 2025
Source of the Lean formalization adapted here.
▸For your AI
I am proving a theorem in Lean 4 and submitting it to Conjectura.
## Problem NT007 — Is the totient series irrational?
Sum φ(n)/2ⁿ over all n, where φ counts integers coprime to n. Is the result irrational?
## Environment (fixed — do not assume anything newer)
- Lean toolchain: `leanprover/lean4:v4.33.0-rc1`
- Mathlib: `v4.33.0-rc1`
If a lemma you want does not exist in that Mathlib, prove it inline instead of
importing something newer.
## The exact statement I must prove
```lean
theorem solution : Irrational (∑' n : ℕ, (φ n : ℝ) / (2 ^ n)) := by
sorry
```
## The file I submit
```lean
import Conjectura.Problems.NT007.Statement
namespace Submission
theorem solution : Conjectura.NT007.goal := by
sorry
end Submission
```
## The Lean definitions of every term in this problem
These are the actual definitions your proof will be checked against. Do not
substitute your own version of any of them.
### Irrational
def
```lean
Irrational
```
Defined in Mathlib.
## Rules — submissions violating these are rejected automatically
1. **Do not change the name or type of `solution`.** It must satisfy the
statement above exactly.
2. **Do not redefine or shadow anything from the problem's Statement module.**
Declaring your own `goal`, or redefining a name it depends on, produces a
proof of a *different* statement and is rejected. This is the single most
common rejection.
3. **No `sorry`** anywhere, including in helper lemmas. It surfaces as the
axiom `sorryAx` and is detected transitively through imports.
4. **No `native_decide`** — it surfaces as `Lean.ofReduceBool` and is not
accepted, because it trusts compiled code rather than the kernel.
5. Only these axioms are permitted: `propext`, `Classical.choice`,
`Quot.sound`.
6. Follow Mathlib style: hypotheses left of the colon, explicit types,
`snake_case` theorem names, `UpperCamelCase` types.
## What I want from you
Here is my argument in informal mathematics:
> [PASTE YOUR PROOF SKETCH HERE]
Turn it into Lean 4 that compiles under the environment above and satisfies the
statement exactly. Where you are unsure a lemma exists in this Mathlib version,
say so explicitly rather than guessing a name.Submissions are not open yet
Conjectura is in beta. You can read every statement, every definition and the Lean behind them, and download the exact files the checker uses — but proofs are not being accepted yet.
The reason is a deliberate order of operations. Accepting a proof means running a stranger’s code and standing behind a verdict, and no statement here yet carries a researcher’s name. A machine-checked answer to a question nobody has vouched for is worth very little, so the vouching comes first.
The English write-ups are also switched off during the beta. Nothing on this page is generated by a model.
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