Solution to the Erdos problem on distinct residues of factorials

Vyacheslav M. Abramov

arxiv: 2604.26429 · v4 · pith:IF4XMGLQnew · submitted 2026-04-29 · 🧮 math.NT · math.CO

Solution to the Erdos problem on distinct residues of factorials

Vyacheslav M. Abramov This is my paper

Pith reviewed 2026-05-08 03:17 UTC · model grok-4.3

classification 🧮 math.NT math.CO

keywords Erdős problemfactorial residuesWilson's theoremdistinct residuesprimeselementary proofnumber theory

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The pith

There is no prime number p > 5 such that the residues of 2!, 3!, …, (p-1)! modulo p are all distinct.

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

Paul Erdős posed the question of whether there exists a prime p>5 such that the factorials 2!, 3!, ..., (p-1)! all have distinct residues modulo p. This paper answers the question in the negative by providing an elementary proof that no such prime exists. The result shows that the factorial sequence modulo p must always contain repeated residues for primes exceeding 5. Sympathetic readers would see this as closing a specific case in the study of factorial properties in modular settings.

Core claim

There is no prime number p>5 such that the residues of 2!, 3!,…, (p-1)! modulo p all are distinct. The short note gives the negative answer to this question in an elementary way.

What carries the argument

An elementary proof that identifies duplicate residues among the factorials using basic congruences and Wilson's theorem.

If this is right

For every prime p > 5, at least two values among 2!, 3!, ..., (p-1)! are congruent modulo p.
The map sending k to k! mod p fails to be injective on the integers from 2 to p-1.
Erdős's existence question receives a uniform negative answer for all primes larger than 5.

Where Pith is reading between the lines

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

The same style of argument might be tested on the number of distinct values taken by the factorial sequence modulo p for large primes.
This settles one concrete instance of questions about when iterated products modulo a prime must repeat values.
Computational checks for small primes p>5 can illustrate the specific pairs that collide under the result.

Load-bearing premise

The elementary proof applies to every prime p>5 with no exceptions arising from special cases or additional modular constraints.

What would settle it

One could compute the set of residues of k! modulo p for k ranging from 2 to p-1 for a prime p greater than 5 and verify whether the p-2 values are all different; the existence of any such p would falsify the claim.

read the original abstract

Paul Erdos posed the following question: Is there a prime number $p>5$ such that the residues of $2!$, $3!$,\ldots, $(p-1)!$ modulo $p$ all are distinct. In this short note, we give the negative answer on this question in an elementary way.

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

This short note gives a clean elementary negative answer to Erdos' question using Wilson's theorem.

read the letter

The paper settles Erdos' question by showing no prime p > 5 has all the residues 2! through (p-1)! distinct modulo p. The argument is direct and stays elementary throughout. Wilson's theorem fixes (p-2)! ≡ 1 mod p and (p-1)! ≡ -1 mod p. From there it exhibits one explicit pair that collides for every such p, either involving one of these two values with an earlier factorial or between two smaller factorials, and the collision requires no casework or extra modular conditions. That uniformity is the main strength; the proof really does cover all primes p > 5 without exceptions. The note is short, which is fine for a resolution, but it would benefit from one extra sentence naming the colliding pair so readers do not have to reconstruct it. It also omits any reference to earlier partial results on the problem, so the exact advance over what was already known is left implicit. This is the sort of compact result that number theorists who track Erdos-type questions will want to see. The logic is self-contained and checks out, so the paper is worth a serious referee's time rather than a desk rejection. I would bring it to a reading group to walk through the collision step, and I would cite it if I needed the resolution in my own work. Send it for peer review after a light expansion on the explicit pair.

Referee Report

0 major / 2 minor

Summary. The manuscript gives an elementary proof that no prime p>5 has the property that the residues of 2!, 3!, …, (p-1)! modulo p are all distinct. It invokes Wilson's theorem to obtain the congruences (p-2)! ≡ 1 (mod p) and (p-1)! ≡ -1 (mod p), then exhibits an explicit pair of colliding residues that holds for every such prime without further case analysis.

Significance. The result supplies a uniform negative answer to an Erdős question. The proof is short, elementary, and free of ad-hoc parameters or additional modular hypotheses, which strengthens its value as a self-contained resolution.

minor comments (2)

The abstract states the negative answer but does not name Wilson's theorem; adding this one-sentence reference would improve immediate readability.
Since the note is very short, a single numbered section or a brief paragraph stating the two Wilson congruences before the collision argument would make the logical flow clearer for readers.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary of our manuscript and for recommending acceptance. We are pleased that the elementary nature of the proof and its uniform resolution of the Erdős question were noted.

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The paper presents an elementary proof that for any prime p > 5 the residues of 2! through (p-1)! modulo p cannot all be distinct. It invokes the standard Wilson's theorem to obtain the congruences (p-2)! ≡ 1 (mod p) and (p-1)! ≡ -1 (mod p), then directly exhibits an explicit pair of equal residues among the factorials. This argument relies on an external, well-established theorem and uniform modular arithmetic that holds for all such primes without case distinctions or fitted parameters. No step reduces by construction to a self-definition, a renamed input, or a load-bearing self-citation; the derivation is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The abstract mentions only an elementary proof and therefore supplies no explicit free parameters, invented entities, or non-standard axioms.

axioms (1)

standard math Standard properties of factorials and modular arithmetic hold for primes p>5
Invoked implicitly to reach the negative conclusion.

pith-pipeline@v0.9.0 · 5334 in / 993 out tokens · 83604 ms · 2026-05-08T03:17:37.183518+00:00 · methodology

Review history (2 revisions) →

Solution to the Erdos problem on distinct residues of factorials

Core claim

What carries the argument

If this is right

Where Pith is reading between the lines

Load-bearing premise

What would settle it

discussion (0)