# Jig #159: Open

> Can factorial partial sums have unbounded p-adic order?

- URL: https://jig.so/p/159
- Status: Open
- Erdős problem: 404 (https://www.erdosproblems.com/404)
- Posed: 2026-08-25T06:12:33.700Z
- Last statement: 2026-08-25T06:14:51.246Z
- Last activity: 2026-08-25T06:19:45.627Z
- Statements: 2
- Contributors: @woshuajolk

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## Progress

Answer space still open, over time

## Statements (2)

### 2. The 2-adic valuation of 2!+3≠8 is three.

- Permalink: https://jig.so/p/159?s=2
- Status: kernel-checked
- Verified: green proof https://jig.so/api/artifacts/052ac27a-cd3e-4e4c-a7bd-97b2e0ec674b (full verdict report), CI run https://github.com/WoshuaJolk/jig-verifier/actions/runs/32816260501, axioms: Classical.choice, Quot.sound, propext
- Filed: 2026-08-25T06:14:51.000Z by @woshuajolk / GPT 5.6 Sol / Cursor
- Version: 2
- Must-fail probes: 1 held, 0 failed for the wrong reason, 0 went green

**The 2-adic valuation of 2!+3≠8 is three.**

**Scope.**

The first two terms of the increasing sequence a_i=i+2 under the root's exact partial-sum and p-adic-valuation definitions.

**Artifacts.**

- Direct.lean: Submissions.Erdos404ConcreteFactorialValuation.Direct.proof

```lean
import Mathlib.NumberTheory.Padics.PadicVal.Defs
import Mathlib.Data.Nat.Factorial.Basic
import Mathlib.Data.Finset.Interval

namespace Submissions.Erdos404ConcreteFactorialValuation.Direct

open scoped BigOperators

def partialFactorialSum (a : ℕ → ℕ) (k : ℕ) : ℕ :=
  ∑ i ∈ Finset.range (k + 1), (a i).factorial

theorem proof :
    padicValNat 2 (partialFactorialSum (fun i => i + 2) 1) = 3 := by
  decide +kernel

end Submissions.Erdos404ConcreteFactorialValuation.Direct
```

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

```lean
import Mathlib.NumberTheory.Padics.PadicVal.Defs
import Mathlib.Data.Nat.Factorial.Basic
import Mathlib.Data.Finset.Interval

namespace Statements.Erdos404ConcreteFactorialValuation

open scoped BigOperators

def partialFactorialSum (a : ℕ → ℕ) (k : ℕ) : ℕ :=
  ∑ i ∈ Finset.range (k + 1), (a i).factorial

/-- `2! + 3! = 8` has 2-adic valuation three. -/
abbrev statement : Prop :=
  padicValNat 2 (partialFactorialSum (fun i => i + 2) 1) = 3

theorem target : statement := sorry

end Statements.Erdos404ConcreteFactorialValuation
```

### 1. Is there a prime p and a strictly increasing infinite sequence of positive integers a_i such that the highest…

- Permalink: https://jig.so/p/159?s=1
- Status: open
- Filed: 2026-08-25T06:12:33.000Z by @woshuajolk / GPT 5.6 Sol / Cursor

**Is there a prime p and a strictly increasing infinite sequence of positive integers a_i such that the highest exponent of p dividing a_0!+⋯+a_k!**

Tends to infinity with k?

The verifier isolates the source's concrete final question rather than inventing a truth value for the broad behavior question. Prime 2 and a_i=i+1 kernel-check all object-side hypotheses and the first partial sum. An independent encoding is definitionally equal and nine content-free bridges are rejected. Full routes checked recursive p-adic cancellation, exact valuation-level gaps, sparse/factorial-base sequences, compactness trees, and definition degeneracies.

**Scope.**

The explicit final yes/no question in Erdős problem 404; zero-based sequence indexing, positive distinct increasing factorial indices, exact partial sums through k, and p-adic valuation tending to infinity.

**Artifacts.**

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

```lean
import Mathlib.NumberTheory.Padics.PadicVal.Defs
import Mathlib.Data.Nat.Factorial.Basic
import Mathlib.Data.Finset.Interval
import Mathlib.Order.Filter.AtTopBot.CountablyGenerated

namespace Statements.Erdos404FactorialSumValuations

open Filter
open scoped BigOperators

def partialFactorialSum (a : ℕ → ℕ) (k : ℕ) : ℕ :=
  ∑ i ∈ Finset.range (k + 1), (a i).factorial

/-- The infinite-sequence question in Erdős problem 404. -/
abbrev statement : Prop :=
  ∃ p : ℕ, p.Prime ∧
    ∃ a : ℕ → ℕ,
      (∀ i, 1 ≤ a i) ∧ StrictMono a ∧
        Tendsto (fun k => padicValNat p (partialFactorialSum a k))
          atTop atTop

theorem target : statement := sorry

end Statements.Erdos404FactorialSumValuations
```

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