Legendre's formula and $p$-adic analysis

Gennady Eremin

arxiv: 1907.11902 · v1 · pith:MVUZDZN7new · submitted 2019-07-27 · 🧮 math.NT · math.CO

Legendre's formula and p-adic analysis

Gennady Eremin This is my paper

Pith reviewed 2026-05-24 14:37 UTC · model grok-4.3

classification 🧮 math.NT math.CO

keywords p-adic valuationLegendre formulap-adic weightfactorialincrementsarithmetic structurenumber theory

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The pith

Increments of p-adic valuation and weight obey their own arithmetic independent of Legendre summation formulas.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

Legendre's formula computes the exponent of prime p in n! either as a sum of floor(n/p^k) or as (n minus the base-p digit sum) divided by (p-1). The paper examines how both expressions change when the argument increases, treating those changes as increments. It proposes defining addition and other operations directly on the increments of valuation and weight. This structure would let one track valuations of factorials, binomials, and Catalan numbers by manipulating increments alone. A reader would care because the approach separates incremental changes from the original floor or digit-sum calculations.

Core claim

The paper examines the relationship between the p-adic valuation and p-adic weight and considers their increments. The arithmetic of the p-adic increments is proposed, allowing the increments of the valuation and the weight to be defined and manipulated independently of the original summation formulas.

What carries the argument

The arithmetic of the p-adic increments, which equips changes in valuation and weight with addition and other operations that act separately from floor sums or digit sums.

If this is right

Valuations of successive factorials can be obtained by adding valuation increments rather than recomputing floor sums.
The same increment arithmetic applies to the alternative digit-sum expression for the valuation.
Binomial coefficients, Catalan numbers, and odd factorials admit valuation tracking through their increment rules.
The relationship between valuation and weight reduces to arithmetic relations on their respective increments.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

One could define explicit tables for increment addition by examining base-p carry patterns when n increases.
The structure might allow closed-form tracking of valuation sequences without iteration over powers of p.
If consistent, the arithmetic could apply to valuations in other p-adic combinatorial settings such as generating functions.

Load-bearing premise

The increments of the valuation and the weight admit a coherent arithmetic structure that can be defined and manipulated independently of the original summation formulas.

What would settle it

A specific n and p where the result of applying the proposed increment arithmetic to reach v_p((n+1)!) differs from the value obtained by direct substitution into either form of Legendre's formula.

read the original abstract

In number theory, we know Legendre's formula $ v_p(n!) = \sum_{k \ge 1} \lfloor \frac{n}{p^k} \rfloor $, which calculates the $p$-adic valuation of the factorial, i.e. the exponent of the greatest power of a prime $p$ that divides $n!$. There is also the second (or alternative) equality $ v_p (n!) = \frac{n-s_p(n)}{p-1} $ where $s_p(n)$ is the $p$-adic weight of $n$ or the sum of digits of $n$ in base $p$. Both kinds of Legendre's formula allow us to determine valuations of the natural number, the odd factorial, binomial coefficients, Catalan numbers, and other combinatorial objects. The article examines the relationship between the $p$-adic valuation and $p$-adic weight and considers their increments. The arithmetic of the $p$-adic increments is proposed.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper restates the two equivalent Legendre formulas and proposes an arithmetic on their increments, but supplies no explicit operations or derivations that would make the arithmetic independent of the known linear relation.

read the letter

The main thing here is that the paper recalls the standard sum formula for v_p(n!) and the equivalent form using the base-p digit sum s_p(n), then says it will look at increments of both and proposes an arithmetic for those increments. That is the entire contribution on offer. It correctly notes the usual applications to binomial coefficients and Catalan numbers, but those are already well-known uses of Legendre's formula. The writing is straightforward and the recall of the two expressions is accurate. Beyond that, nothing new is derived or proved. The stress-test point lands: the identity v_p(n!) = (n - s_p(n))/(p-1) directly ties the valuation and the weight, so any increment in one is rigidly determined by the other once you fix the base-p carries. The abstract claims a coherent arithmetic that can be manipulated independently of the original summation formulas, yet the text gives no definitions, no axioms, and no worked examples that would escape this dependence. Without those details it is impossible to tell whether the proposal adds anything beyond re-labeling quantities already fixed by the identity. This is the sort of note that might interest someone already working on p-adic valuations who wants to see whether the increment idea can be made to do new work, but the current write-up does not demonstrate that it can. I would not bring it to a reading group, would not cite it, and would not send it to a serious referee; the central claim about an independent arithmetic is not secured by any explicit construction.

Referee Report

1 major / 0 minor

Summary. The paper recalls Legendre's two equivalent formulas for the p-adic valuation v_p(n!): the summation form sum_{k>=1} floor(n/p^k) and the closed form (n - s_p(n))/(p-1) where s_p(n) is the sum of base-p digits. It examines the relationship between the valuation and the p-adic weight, considers their increments, and proposes an arithmetic of the p-adic increments.

Significance. If a coherent, independent arithmetic on increments were established, it could supply new tools for analyzing valuations of factorials, binomial coefficients, and Catalan numbers. However, the manuscript supplies no explicit operations, axioms, or derivations, so no such significance can be assessed from the given text.

major comments (1)

[Abstract] Abstract, final sentence: the claim that 'the arithmetic of the p-adic increments is proposed' with the implication that it can be 'defined and manipulated independently of the summation formulas' is unsupported. The identity v_p(n!) = (n - s_p(n))/(p-1) directly links any increment in valuation to the corresponding change in s_p(n) via the base-p carry structure; without explicit operations or axioms that escape this relation, the independence cannot be verified.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful review of our manuscript. We respond to the major comment on the abstract below.

read point-by-point responses

Referee: [Abstract] Abstract, final sentence: the claim that 'the arithmetic of the p-adic increments is proposed' with the implication that it can be 'defined and manipulated independently of the summation formulas' is unsupported. The identity v_p(n!) = (n - s_p(n))/(p-1) directly links any increment in valuation to the corresponding change in s_p(n) via the base-p carry structure; without explicit operations or axioms that escape this relation, the independence cannot be verified.

Authors: We agree that the manuscript does not furnish explicit operations, axioms, or derivations establishing an arithmetic of p-adic increments that can be manipulated independently of the summation formulas or the closed-form identity. The text recalls Legendre's formulas, examines the relationship between v_p(n!) and s_p(n), and considers increments arising from that relationship, but does not develop a separate axiomatic framework. The abstract's wording therefore overstates the scope of the contribution. We will revise the abstract (and, if appropriate, the introduction) to describe the paper's actual content without claiming an independent arithmetic. revision: yes

Circularity Check

0 steps flagged

No circularity exhibited; proposal of increment arithmetic not shown to reduce to inputs

full rationale

The provided text consists only of the abstract, which states that the paper examines the relationship between valuation and weight, considers their increments, and proposes an arithmetic of p-adic increments. No explicit equations, derivations, or definitions of the proposed arithmetic appear. Without any load-bearing step that can be quoted and shown to reduce by construction to the known Legendre identities or to a fitted input, no circularity of any enumerated kind is identifiable. The derivation chain is therefore treated as self-contained on the available evidence.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Only the abstract is available, so no free parameters, axioms, or invented entities can be extracted or audited.

pith-pipeline@v0.9.0 · 5689 in / 950 out tokens · 19913 ms · 2026-05-24T14:37:36.579969+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/ArithmeticFromLogic.lean LogicNat recovery; embed_injective; no connection unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

The arithmetic of the p-adic increments is proposed... Δv_p(n!)=(1-Δs_p(n))/(p-1)... Δs_p(n)=1-(p-1)v_p(n+1)
IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel; J-cost functional equation unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

v_p(n!)=(n-s_p(n))/(p-1) ... increments rigidly linked

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.