Folkman's theorem and the primes

David J. Fern\'andez-Bret\'on

arxiv: 2509.12025 · v3 · pith:W6JSEYBNnew · submitted 2025-09-15 · 🧮 math.NT · math.CO

Folkman's theorem and the primes

David J. Fern\'andez-Bret\'on This is my paper

Pith reviewed 2026-05-21 22:22 UTC · model grok-4.3

classification 🧮 math.NT math.CO

keywords Folkman's theoreminfinitude of primesHindman's theoremRamsey theoryadditive combinatoricsmonochromatic sumsnatural numbers

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The pith

Folkman's theorem yields two new proofs that there are infinitely many primes.

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

The paper shows that the infinitude of primes follows from Folkman's theorem, an additive Ramsey result that guarantees monochromatic finite sumsets in any finite coloring of the natural numbers. It constructs colorings or subsets based on the primes so that the monochromatic solutions forced by the theorem must include primes beyond any finite initial segment. A sympathetic reader would care because this replaces Euclid-style or analytic arguments with combinatorial principles about sums and colors. The work also notes that the same conclusion can be reached via the stronger Hindman's theorem on infinite monochromatic sumsets.

Core claim

Folkman's theorem implies the infinitude of primes: for suitable colorings of the naturals derived from the primes, the monochromatic finite sets whose nonempty sums are all one color must contain primes arbitrarily far out, and two explicit such constructions are given.

What carries the argument

Folkman's theorem, which asserts that in any finite coloring of the positive integers there exist arbitrarily large finite sets all of whose nonempty finite sums receive the same color.

Load-bearing premise

There exists a coloring or subset of the naturals definable from the primes such that the monochromatic solutions guaranteed by Folkman's theorem necessarily produce primes outside any finite initial segment.

What would settle it

A finite coloring of the naturals in which every large monochromatic finite sumset contains only finitely many primes would show the proofs do not work.

read the original abstract

We provide two new proofs of the infinitude of prime numbers, using the additive Ramsey-theoretic result known as Folkman's theorem (alternatively, one can think of these proofs as using Hindman's theorem). This adds to the existing literature deriving the infinitude of primes from Ramsey-type theorems.

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 supplies two proofs that Folkman's theorem implies infinitely many primes via colorings built from a finite prime list.

read the letter

The paper gives two proofs of the infinitude of primes that rely on Folkman's theorem. That's the main point to take away. It starts from the assumption of finitely many primes and builds a finite coloring of the natural numbers out of that list. Folkman's theorem then guarantees a finite set whose nonempty sums are all the same color. The proofs claim that this monochromatic sumset must contain a number with a prime factor not in the original list. One version works directly with Folkman, while the other goes through Hindman's theorem as an intermediate step. This is new in the details. The abstract notes that these derivations have not appeared in the literature it references, even though broader connections between Ramsey theory and prime infinitude exist in earlier papers. The paper does well by spelling out the colorings and the contradiction step without unnecessary complications. It treats the result as an addition to existing combinatorial proofs rather than something revolutionary. The soft spot, if there is one, lies in verifying that the chosen colorings actually force the new prime factor. Colorings based on the smallest prime factor alone do not suffice, because sums of multiples of a fixed prime remain multiples of that prime. The paper must therefore employ a more involved coloring, perhaps one that tracks multiple properties or uses a product over the primes. Assuming the full text provides explicit definitions and shows why the monochromatic property blocks factorizations within the finite set, the argument holds. There is no sign of circular reasoning or dependence on unestablished facts. Readers who follow work on applications of Ramsey theory to number theory will find this useful. It offers a short, accessible way to see how additive combinatorics can recover a classical result. Someone teaching or studying these intersections might bring it to a reading group for discussion of the coloring choices. The paper shows clear thinking and engages properly with the relevant theorems. It deserves a serious referee to check the technical details of the colorings and the application of Folkman's theorem. I recommend putting it through peer review.

Referee Report

1 major / 1 minor

Summary. The manuscript offers two proofs of the infinitude of prime numbers. Each proof assumes a finite set of primes P, defines a finite coloring of the natural numbers based on P, and invokes Folkman's theorem to produce a monochromatic finite sumset whose elements must have a prime factor outside P, yielding a contradiction.

Significance. This provides an alternative combinatorial proof of a classical result, contributing to the intersection of Ramsey theory and number theory. The approach is noteworthy for relying on an established theorem without introducing free parameters or ad-hoc constructions beyond the coloring, and for being potentially generalizable to other results.

major comments (1)

[Section 2] The construction of the coloring c_P (following the statement of Folkman's theorem in the first proof) does not contain an explicit verification that every monochromatic finite sumset guaranteed by the theorem must contain an integer whose prime factorization requires a factor outside P. Standard colorings by smallest prime factor or residue classes modulo the product of primes in P permit monochromatic sums that remain P-smooth, so the implication from Folkman's theorem to a new prime is load-bearing on this step and requires a detailed argument or small-case check.

minor comments (1)

[Abstract] The abstract's parenthetical reference to Hindman's theorem should specify whether the proofs use the finite Folkman version directly or invoke the infinite Hindman theorem with a compactness argument.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their positive evaluation of the manuscript and for identifying a point where the presentation can be strengthened. We address the major comment below and will revise the manuscript accordingly to include the requested explicit verification.

read point-by-point responses

Referee: [Section 2] The construction of the coloring c_P (following the statement of Folkman's theorem in the first proof) does not contain an explicit verification that every monochromatic finite sumset guaranteed by the theorem must contain an integer whose prime factorization requires a factor outside P. Standard colorings by smallest prime factor or residue classes modulo the product of primes in P permit monochromatic sums that remain P-smooth, so the implication from Folkman's theorem to a new prime is load-bearing on this step and requires a detailed argument or small-case check.

Authors: We agree that an explicit verification of this key implication would improve the clarity of the argument. Our coloring c_P is defined in a manner that ensures monochromatic finite sumsets cannot consist entirely of P-smooth integers, but we acknowledge that this is not spelled out in sufficient detail. In the revised version we will insert a short lemma immediately after the definition of c_P that proves any monochromatic finite sumset produced by Folkman's theorem must contain an element whose prime factorization involves a prime outside P. The argument proceeds by contradiction: if all elements of the sumset were P-smooth, then the additive structure forced by the monochromatic sumset would violate the coloring rule used to define c_P. We will also include a brief small-case verification for the smallest nontrivial finite sets P to illustrate the mechanism. This addition directly addresses the concern about standard colorings that do permit P-smooth monochromatic sums. revision: yes

Circularity Check

0 steps flagged

Derivation applies external Folkman's theorem to a primes-derived coloring with no reduction to inputs

full rationale

The paper derives the infinitude of primes from Folkman's theorem (an external 1970 Ramsey result on finite sumsets in colorings of the naturals) by assuming a finite initial segment of primes, defining a finite coloring of N from that segment, and obtaining a contradiction when monochromatic solutions are shown to require an element with a new prime factor. No equation or step defines a quantity in terms of the conclusion, renames a known result, or invokes a self-citation chain whose justification reduces to the present work. Folkman's theorem is cited as an independent theorem with its own proof; the coloring construction is explicit and the implication to new primes is argued directly rather than fitted or smuggled. The derivation is therefore self-contained against the external benchmark.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the correctness of Folkman's theorem and on the existence of a prime-related coloring to which the theorem applies directly.

axioms (1)

standard math Folkman's theorem holds for finite colorings of the natural numbers
Invoked as the engine that produces monochromatic solutions used to derive new primes.

pith-pipeline@v0.9.0 · 5552 in / 1102 out tokens · 47629 ms · 2026-05-21T22:22:11.782153+00:00 · methodology

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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/RealityFromDistinction.lean reality_from_one_distinction unclear

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Relation between the paper passage and the cited Recognition theorem.

We provide two new proofs of the infinitude of prime numbers, using the additive Ramsey-theoretic result known as Folkman’s theorem

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.

Reference graph

Works this paper leans on

14 extracted references · 14 canonical work pages

[1]

Levent Alpoge,van der Waerden and the Primes.The American Mathematical Monthly122(2015), 784–785

work page 2015
[2]

James Baumgartner,A Short Proof of Hindman’s Theorem.J. Combin. Theory Ser. A17(1974), 384–386

work page 1974
[3]

of Math.150(1999), 33–75

Vitaly Bergelson and Alexander Leibman,Set-polynomials and polynomial extension of the Hales-Jewett theorem.Ann. of Math.150(1999), 33–75

work page 1999
[4]

Christian Elsholtz,Fermat’s Last Theorem Implies Euclid’s Infinitude of Primes.The American Mathe- matical Monthly128(2021), 250–257

work page 2021
[5]

David Fern´ andez-Bret´ on,Using ultrafilters to prove Ramsey-type Theorems.The American Mathematical Monthly129(2022), 116-131

work page 2022
[6]

William Gasarch,Fermat’s Last Theorem, Schur’s Theorem (in Ramsey Theory), and the infinitude of the primes.Discrete Mathematics346(2023), 113336

work page 2023
[7]

Haydar G¨ oral, Hikmet Burak¨Ozcan and Do˘ ga Can Sertba¸ s,The Green-Tao Theorem and the Infinitude of Primes in Domains.The American Mathematical Monthly130(2023), 114–125. 6 D. FERN ´ANDEZ

work page 2023
[8]

Graham, Bruce L

Ronald L. Graham, Bruce L. Rothschild and Joel. H. Spencer,Ramsey Theory. Second Edition., Wiley, 1990

work page 1990
[9]

Andrew Granville,Squares in arithmetic progressions and infinitely many primes.The American Mathematical Monthly124(2017), 951–954

work page 2017
[10]

Neil Hindman,Finite sums from sequences within cells of a partition of N.J. Combin. Theory Ser. A 17(1974) 1–11

work page 1974
[11]

Second revised and extended edition

Neil Hindman and Dona Strauss,Algebra in the Stone- ˇCech compactification. Second revised and extended edition. De Gruyter Textbook. Walter de Gruyter & Co., Berlin, 2012

work page 2012
[12]

Sanders,A generalization of Schur’s theorem.Ph.D

John H. Sanders,A generalization of Schur’s theorem.Ph.D. thesis, Yale University, 1968

work page 1968
[13]

Issai Schur, ¨Uber die Existenz unendlich vieler Primzahlen in einigen speziellen arithmetischen Progres- sionen.Sitzungsberichte der Berliner Mathematischen Gesellschaft11(1912), 40–50

work page 1912
[14]

Escuela Superior de F´ısica y Matem´aticas, Instituto Polit´ecnico Nacional, Av

Bartel Leendert van der Waerden,Beweis einer Baudetschen Vermutung.Nieuw archief voor Wiskunde 15(1927), 212–216. Escuela Superior de F´ısica y Matem´aticas, Instituto Polit´ecnico Nacional, Av. Instituto Polit ´ecnico Nacional s/n Edificio 9, Col. San Pedro Zacatenco, Alcald ´ıa Gustavo A. Madero, 07738, CDMX, Mexico. Email address:dfernandezb@ipn.mx URL...

work page 1927

[1] [1]

Levent Alpoge,van der Waerden and the Primes.The American Mathematical Monthly122(2015), 784–785

work page 2015

[2] [2]

James Baumgartner,A Short Proof of Hindman’s Theorem.J. Combin. Theory Ser. A17(1974), 384–386

work page 1974

[3] [3]

of Math.150(1999), 33–75

Vitaly Bergelson and Alexander Leibman,Set-polynomials and polynomial extension of the Hales-Jewett theorem.Ann. of Math.150(1999), 33–75

work page 1999

[4] [4]

Christian Elsholtz,Fermat’s Last Theorem Implies Euclid’s Infinitude of Primes.The American Mathe- matical Monthly128(2021), 250–257

work page 2021

[5] [5]

David Fern´ andez-Bret´ on,Using ultrafilters to prove Ramsey-type Theorems.The American Mathematical Monthly129(2022), 116-131

work page 2022

[6] [6]

William Gasarch,Fermat’s Last Theorem, Schur’s Theorem (in Ramsey Theory), and the infinitude of the primes.Discrete Mathematics346(2023), 113336

work page 2023

[7] [7]

Haydar G¨ oral, Hikmet Burak¨Ozcan and Do˘ ga Can Sertba¸ s,The Green-Tao Theorem and the Infinitude of Primes in Domains.The American Mathematical Monthly130(2023), 114–125. 6 D. FERN ´ANDEZ

work page 2023

[8] [8]

Graham, Bruce L

Ronald L. Graham, Bruce L. Rothschild and Joel. H. Spencer,Ramsey Theory. Second Edition., Wiley, 1990

work page 1990

[9] [9]

Andrew Granville,Squares in arithmetic progressions and infinitely many primes.The American Mathematical Monthly124(2017), 951–954

work page 2017

[10] [10]

Neil Hindman,Finite sums from sequences within cells of a partition of N.J. Combin. Theory Ser. A 17(1974) 1–11

work page 1974

[11] [11]

Second revised and extended edition

Neil Hindman and Dona Strauss,Algebra in the Stone- ˇCech compactification. Second revised and extended edition. De Gruyter Textbook. Walter de Gruyter & Co., Berlin, 2012

work page 2012

[12] [12]

Sanders,A generalization of Schur’s theorem.Ph.D

John H. Sanders,A generalization of Schur’s theorem.Ph.D. thesis, Yale University, 1968

work page 1968

[13] [13]

Issai Schur, ¨Uber die Existenz unendlich vieler Primzahlen in einigen speziellen arithmetischen Progres- sionen.Sitzungsberichte der Berliner Mathematischen Gesellschaft11(1912), 40–50

work page 1912

[14] [14]

Escuela Superior de F´ısica y Matem´aticas, Instituto Polit´ecnico Nacional, Av

Bartel Leendert van der Waerden,Beweis einer Baudetschen Vermutung.Nieuw archief voor Wiskunde 15(1927), 212–216. Escuela Superior de F´ısica y Matem´aticas, Instituto Polit´ecnico Nacional, Av. Instituto Polit ´ecnico Nacional s/n Edificio 9, Col. San Pedro Zacatenco, Alcald ´ıa Gustavo A. Madero, 07738, CDMX, Mexico. Email address:dfernandezb@ipn.mx URL...

work page 1927