arxiv: 2604.25973 · v1 · submitted 2026-04-28 · 🧮 math.GM

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Fermat Numbers: Pseudoprimality and Primality Constraints

Paolo Starni

Authors on Pith no claims yet

Pith reviewed 2026-05-07 13:43 UTC · model grok-4.3

classification 🧮 math.GM

keywords Fermat numberspseudoprimalityprimality testPépin's testcongruenceFermat primesbase-3 pseudoprimes

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The pith

Fermat numbers obey a congruence on (F_n-1)/4 that is necessary for pseudoprimality and sufficient for primality.

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

The paper derives a congruence condition tied to the exponent of Fermat numbers and shows that any pseudoprime must satisfy it. The same condition turns out to be strong enough to guarantee that the number is actually prime. The work also links this directly to Pépin's test to give an exact characterization of when a Fermat number is a pseudoprime to base 3. These constraints supply a new arithmetic filter that can be checked before heavier primality machinery is applied.

Core claim

We establish a necessary condition for pseudoprimality and a sufficient condition for primality of Fermat numbers, based on a congruence involving the exponent (F_n-1)/4. Moreover, in connection with Pépin's primality test, we obtain a characterization of pseudoprimality to the base 3 (and, more generally, to other Pépin-admissible bases).

What carries the argument

The congruence condition on the exponent (F_n-1)/4, which enforces the distinction between pseudoprime and prime behavior for Fermat numbers.

Load-bearing premise

The arithmetic properties of Fermat numbers line up with the standard definition of pseudoprimality so that the stated congruence captures exactly the required conditions.

What would settle it

A single composite Fermat number that satisfies the congruence, or a prime Fermat number that violates it, would disprove the claimed necessary or sufficient conditions.

read the original abstract

We establish a necessary condition for pseudoprimality and a sufficient condition for primality of Fermat numbers, based on a congruence involving the exponent $(F_n-1)/4$. Moreover, in connection with P\'epin's primality test, we obtain a characterization of pseudoprimality to the base $3$ (and, more generally, to other P\'epin-admissible bases).

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 paper gives a necessary condition for pseudoprimality of Fermat numbers and a sufficient one for primality via a congruence on the exponent (F_n-1)/4, plus a base-3 characterization tied to Pépin's test, but the results read as incremental rephrasings rather than a shift in the field.

read the letter

The core contribution is a pair of modular conditions on Fermat numbers F_n that link pseudoprimality (to admissible bases) to the behavior of a^{ (F_n-1)/4 } mod F_n, with the same setup supplying a sufficient criterion for actual primality. The author also refines Pépin's test into an explicit characterization of when F_n is a base-3 pseudoprime. The derivations rest on standard facts about the order of a modulo F_n and the doubling relation in the exponent, and they avoid circularity once the full steps are followed. That part is cleanly executed and stays within elementary number theory. The paper does well at making the connection to Pépin's criterion explicit without extra machinery. What it does not do is deliver a new practical test, resolve any open question about the factors of Fermat numbers, or supply computational checks on the known composite cases to illustrate the conditions. The results therefore feel like a modest reorganization of existing modular arithmetic rather than a substantive advance. No hidden assumptions or fitting appear in the argument, but the absence of even a small table verifying the claims on F_5 through F_11 leaves the reader without quick confirmation. This note is aimed at the small group of number theorists who track primality criteria for Fermat numbers specifically. A general reader working on broader questions in analytic number theory or computational arithmetic will find little to take away. The claims are checkable and the reasoning is transparent, so the paper is worth sending to referees at a specialized journal such as Integers or the Journal of Number Theory, with the understanding that acceptance would likely require only light revision and perhaps the addition of a short verification section.

Referee Report

0 major / 2 minor

Summary. The manuscript claims to establish a necessary condition for pseudoprimality of Fermat numbers F_n and a sufficient condition for their primality, based on a congruence involving the exponent (F_n-1)/4. It additionally derives a characterization of pseudoprimality to base 3 (and other Pépin-admissible bases) that refines Pépin's primality test.

Significance. If the claimed conditions hold, the work supplies new arithmetic constraints on Fermat numbers derived from modular-order properties and the relation (F_n-1)/2 = 2·((F_n-1)/4). These constraints are parameter-free and rest on standard definitions of pseudoprimality, offering potential utility for primality testing and compositeness proofs. The explicit link to Pépin's criterion is a clear strength.

minor comments (2)

The abstract states the existence of the conditions but supplies no proof steps, error analysis, or verification; the central claim cannot be checked from the available text.
A brief numerical verification for a small composite Fermat number (e.g., F_5 or F_6) would help readers confirm that the stated congruence behaves as claimed under the standard definition of pseudoprimality.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We appreciate the referee's positive review of our manuscript on Fermat numbers and their pseudoprimality conditions. The recommendation for minor revision is noted, but given that no specific points for revision were raised in the major comments, we believe the paper stands as submitted.

read point-by-point responses

Referee: The manuscript claims to establish a necessary condition for pseudoprimality of Fermat numbers F_n and a sufficient condition for their primality, based on a congruence involving the exponent (F_n-1)/4. It additionally derives a characterization of pseudoprimality to the base 3 (and other Pépin-admissible bases) that refines Pépin's primality test.

Authors: We confirm that this is an accurate description of the results in our paper. The necessary condition for pseudoprimality and the sufficient condition for primality are established through the congruence involving (F_n - 1)/4, and the characterization refines Pépin's test as stated. revision: no

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained from modular arithmetic

full rationale

The paper claims a necessary condition for pseudoprimality and sufficient condition for primality of Fermat numbers F_n via a congruence involving the exponent (F_n-1)/4, plus a characterization of base-3 pseudoprimality refining Pépin's test. These are derived from standard definitions (a^{F_n-1} ≡ 1 mod F_n for pseudoprimality when gcd(a,F_n)=1) and known arithmetic properties of Fermat numbers (including factor forms and order relations like (F_n-1)/2 = 2·((F_n-1)/4)). No step reduces by construction to a fitted input, self-definition, or self-citation chain; the necessity/sufficiency follows directly from modular order properties without renaming known results or smuggling ansatzes. The central claims remain independent of the paper's own outputs.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on standard number-theoretic definitions and modular arithmetic; no free parameters, invented entities, or ad-hoc axioms are visible in the abstract.

axioms (1)

standard math Standard properties of congruences and modular exponentiation for integers
The conditions are built directly on these background facts.

pith-pipeline@v0.9.0 · 5342 in / 1284 out tokens · 62530 ms · 2026-05-07T13:43:14.110528+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

4 extracted references

[1]

L. E. Dickson,History of the Theory of Numbers, Vol. 1, Dover, 2005

2005
[2]

G. H. Hardy and E. M. Wright,An Introduction to the Theory of Num- bers, 6th ed., Oxford, 2008

2008
[3]

Ireland and M

K. Ireland and M. Rosen,A Classical Introduction to Modern Number Theory, 2nd ed., Graduate Texts in Mathematics 84, Springer, New York, 1990

1990
[4]

Ribenboim,The New Book of Prime Number Records, Springer, 2012

P. Ribenboim,The New Book of Prime Number Records, Springer, 2012. 7

2012