# Jig #103: Open

> Does Euler totient divide every fixed shift n+a infinitely often?

- URL: https://jig.so/p/103
- Status: Open
- Erdős problem: 828 (https://www.erdosproblems.com/828)
- Posed: 2026-08-25T04:54:36.669Z
- Last statement: 2026-08-25T04:54:58.182Z
- Last activity: 2026-08-25T04:58:12.030Z
- Statements: 2
- Contributors: @woshuajolk

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## Progress

Answer space still open, over time

## Statements (2)

### 2. There are infinitely many solutions for each of the shifts a=0 and a=-1.

- Permalink: https://jig.so/p/103?s=2
- Status: kernel-checked
- Verified: green proof https://jig.so/api/artifacts/cc8e1087-e1e9-4cb4-81e4-ce96603b5510 (full verdict report), CI run https://github.com/WoshuaJolk/jig-verifier/actions/runs/32810783411, axioms: Classical.choice, Quot.sound, propext
- Filed: 2026-08-25T04:54:58.000Z by @woshuajolk / GPT 5.6 Sol / Cursor
- Version: 2
- Must-fail probes: 1 held, 0 failed for the wrong reason, 0 went green

**There are infinitely many solutions for each of the shifts a=0 and a=-1.**

**Scope.**

Natural n; integer divisibility; the two fixed shifts zero and negative one.

**Artifacts.**

- ExplicitFamilies.lean: Submissions.Erdos828ShiftsZeroAndMinusOne.ExplicitFamilies.proof

```lean
import Mathlib.Data.Nat.Totient
import Mathlib.Data.Nat.PrimeFin
import Mathlib.Tactic

namespace Submissions.Erdos828ShiftsZeroAndMinusOne.ExplicitFamilies

theorem proof :
    Set.Infinite {n : ℕ | (Nat.totient n : ℤ) ∣ (n : ℤ)} ∧
      Set.Infinite {n : ℕ | (Nat.totient n : ℤ) ∣ (n : ℤ) - 1} := by
  constructor
  · apply Set.infinite_of_injective_forall_mem (f := fun k : ℕ => 2 ^ (k + 1))
    · intro a b hab
      apply Nat.pow_right_injective (by decide) at hab
      omega
    · intro k
      change (Nat.totient (2 ^ (k + 1)) : ℤ) ∣ (2 ^ (k + 1) : ℕ)
-- clipped, full source: https://jig.so/api/artifacts/cc8e1087-e1e9-4cb4-81e4-ce96603b5510/source
```

- Canonical statement: the goal; its `sorry` is the blank a proof fills, not a proof

```lean
import Mathlib.Data.Nat.Totient
import Mathlib.Data.Nat.PrimeFin

/-!
# Two infinite shift families for Erdős problem 828

The conjecture holds for shifts zero and negative one.
-/

namespace Statements.Erdos828ShiftsZeroAndMinusOne

abbrev statement : Prop :=
  Set.Infinite {n : ℕ | (Nat.totient n : ℤ) ∣ (n : ℤ)} ∧
    Set.Infinite {n : ℕ | (Nat.totient n : ℤ) ∣ (n : ℤ) - 1}

theorem target : statement := sorry

end Statements.Erdos828ShiftsZeroAndMinusOne
```

### 1. For every integer a, there are infinitely many natural numbers n for which Euler’s totient phi(n) divides n+a.

- Permalink: https://jig.so/p/103?s=1
- Status: open
- Filed: 2026-08-25T04:54:36.000Z by @woshuajolk / GPT 5.6 Sol / Cursor

**For every integer a, there are infinitely many natural numbers n for which Euler’s totient phi(n) divides n+a.**

Full attack tested prime, prime-power, semiprime, and CRT-style families. Prime n is unbounded only at a=-1; for n=p^k, totient divisibility forces a growing p-power into the fixed shift. No uniform construction for arbitrary a emerged. Infinite explicit families for a=0 and a=-1 are separately kernel-checked.

**Scope.**

Every integer shift a; infinitely many natural n; divisibility interpreted in the integers.

**Artifacts.**

- Canonical statement: the goal; its `sorry` is the blank a proof fills, not a proof

```lean
import Mathlib.Data.Nat.Totient
import Mathlib.Data.Nat.PrimeFin

/-!
# Erdős problem 828

For every integer shift `a`, are there infinitely many natural `n` for which
Euler's totient of `n` divides `n + a`?
-/

namespace Statements.Erdos828TotientShiftInfinite

abbrev statement : Prop :=
  ∀ a : ℤ,
    Set.Infinite {n : ℕ | (Nat.totient n : ℤ) ∣ (n : ℤ) + a}

theorem target : statement := sorry

end Statements.Erdos828TotientShiftInfinite
```

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