---
title: "van der Waerden and the primes"
authors:
  - "Levent Alpoge"
year: 2015
venue: "American Mathematical Monthly"
arxiv: null
doi: "10.4169/amer.math.monthly.122.8.784"
source: "https://doi.org/10.4169/amer.math.monthly.122.8.784"
topics:
  - "number-theory"
seed_rank: 1217
seed_batch: "alpoge-math-2026-08-24"
reviewed: "2026-08-24"
pool: "maths-foundations"
relevance_score: 8
lineage: arithmetic-geometry
cites:
  []
see:
  - "1213-proof-of-a-conjecture-of-stanley-zanello"
---

# van der Waerden and the primes

## One-sentence takeaway

Colour n by the set of primes dividing n and the parities of their valuations; van der Waerden then forbids finitely many primes — a two-page Monthly proof of infinitude of primes.

## Why it's lovely

The tension between additive regularity and unique factorization, in public. Granville wrote a follow-up Monthly piece starting from this note.

## Problem

Euclid, Euler, Furstenberg, ... — another proof there are infinitely many primes.

## Design

If only finitely many primes, the colouring uses finitely many colours, so monochromatic APs exist; valuation inequalities on a+d, a+2d, ... blow up.

## Evidence

Amer. Math. Monthly 122 (2015), no. 8, 784-785. DOI 10.4169/amer.math.monthly.122.8.784. No arXiv (short note). PDF on the Harvard page / alpo.ge.

## Limitations

Expository note, not a research theorem. Still a sourced journal paper — keep.

## Implications for Broadside

The charming undergraduate note. Skip the HCMR quadratic-reciprocity exposition (Dirichlet already had it).

## Bottom line

Mint the Monthly note.

## Links

- DOI: [10.4169/amer.math.monthly.122.8.784](https://doi.org/10.4169/amer.math.monthly.122.8.784)