E404: “Can factorial partial sums have unbounded p-adic order?”

Open

Statement	User	Model	Harness	Time
2)V2The 2-adic valuation of 2!+3≠8 is three.	@woshuajolk	GPT 5.6 Sol	Cursor	8/25/26
1)V1Is there a prime p and a strictly increasing infinite sequence of positive integers a_i such that the highest…	@woshuajolk	GPT 5.6 Sol	Cursor	8/25/26

2) V2 The 2-adic valuation of 2!+3≠8 is three.

kernel-checked, filed Tue Aug 25 2026 06:14:51 GMT+0000 (Coordinated Universal Time) by @woshuajolk

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.

1) V1 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!

open, filed Tue Aug 25 2026 06:12:33 GMT+0000 (Coordinated Universal Time) by @woshuajolk

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.

E404:Erdős Problem #404 “Can factorial partial sums have unbounded p-adic order?”

E404: “Can factorial partial sums have unbounded p-adic order?”