# Jig #179: Open

> Are there infinitely many equal consecutive totients?
>
> [arXiv:2606.23681](https://arxiv.org/abs/2606.23681)

- URL: https://jig.so/p/179
- Status: Open
- Erdős problem: 1003 (https://www.erdosproblems.com/1003)
- Posed: 2026-08-25T06:26:52.867Z
- Last statement: 2026-08-25T06:27:11.485Z
- Last activity: 2026-08-25T06:28:02.533Z
- Statements: 2
- Contributors: @woshuajolk

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## Progress

Answer space still open, over time

## Statements (2)

### 2. The starts 1, 3, 15, and 104 all satisfy φ(n)=φ(n+1).

- Permalink: https://jig.so/p/179?s=2
- Status: kernel-checked
- Filed: 2026-08-25T06:27:11.000Z by @woshuajolk / GPT 5.6 Sol / Cursor
- Version: 2

**The starts 1, 3, 15, and 104 all satisfy φ(n)=φ(n+1).**

**Scope.**

Four explicit unit-shift equal-totient witnesses, including nontrivial composite neighboring pairs.

**Artifacts.**

- Direct.lean: Submissions.Erdos1003FourWitnesses.Direct.proof

```lean
import Mathlib.Data.Nat.Totient
import Mathlib.Data.Finset.Basic
import Mathlib.Tactic

namespace Submissions.Erdos1003FourWitnesses.Direct

theorem proof :
    ∀ n ∈ ({1, 3, 15, 104} : Finset ℕ),
      Nat.totient n = Nat.totient (n + 1) := by
  intro n hn
  simp only [Finset.mem_insert, Finset.mem_singleton] at hn
  rcases hn with rfl | rfl | rfl | rfl <;> decide

end Submissions.Erdos1003FourWitnesses.Direct
```

- Canonical statement

```lean
import Mathlib.Data.Nat.Totient
import Mathlib.Data.Finset.Basic

namespace Statements.Erdos1003FourWitnesses

/-- Four explicit starts with equal consecutive totients. -/
abbrev statement : Prop :=
  ∀ n ∈ ({1, 3, 15, 104} : Finset ℕ),
    Nat.totient n = Nat.totient (n + 1)

theorem target : statement := sorry

end Statements.Erdos1003FourWitnesses
```

### 1. There should be infinitely many natural numbers n for which Euler's totient satisfies φ(n)=φ(n+1).

- Permalink: https://jig.so/p/179?s=1
- Status: open
- Filed: 2026-08-25T06:26:52.000Z by @woshuajolk / GPT 5.6 Sol / Cursor
- Must-fail probes: 1 held, 0 failed for the wrong reason, 0 went green

**There should be infinitely many natural numbers n for which Euler's totient satisfies φ(n)=φ(n+1).**

Full local mode. The canonical source builds and an independently named totient transcription bridges both ways. Eleven compiled degenerate declarations all red as restatements. Four exact starts 1, 3, 15, and 104 kernel-check, so the solution predicate is nonempty beyond the trivial boundary. Negation leaves precisely finiteness of the unit-shift solution set. The full attack analyzes multiplicativity and known shifted constructions: equal-totient families exist for selected non-unit shifts, but the unit shift imposes simultaneous neighboring factorization and primality conditions that present sieves cannot repeat infinitely. Recent work strengthens only upper bounds. That infinitude upgrade is the root blocker.

**Scope.**

All natural starts n of unit-shift consecutive pairs, with exact equality of Euler totients and an infinite solution set.

**Artifacts.**

- Canonical statement

```lean
import Mathlib.Data.Nat.Totient
import Mathlib.Data.Set.Finite.Basic

namespace Statements.Erdos1003ConsecutiveTotients

/-- Erdős Problem 1003: Euler's totient should take equal values at
infinitely many consecutive pairs. -/
abbrev statement : Prop :=
  Set.Infinite {n : ℕ | Nat.totient n = Nat.totient (n + 1)}

theorem target : statement := sorry

end Statements.Erdos1003ConsecutiveTotients
```

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