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Problem 439
Is it true that, in any finite colouring of the integers, there must be two integers of the same colour such that is a square? What about a th power?
Source: erdosproblems.com/439
- Number theory
- Ramsey theory
Status
Solved, ProvedAn accepted solution exists. The statement is true.
Proved. The status-defining source is Khalfalah and Szemerédi's theorem (Combin. Probab. Comput. 15 (2006), no. 1--2, 213--227, published online 3 January 2006, refereed): for every non-constant polynomial with integer coefficients that takes an even value, every finite coloring of the integers has distinct , of the same color with ; the squares are the case and the th powers the case , which takes the even value . The paper is closed access and not held, so its theorem is cited here through the publisher's abstract, the introduction of Sanders's refereed 2020 note (the square case, with the distinctness of and stated), Green and Lindqvist's remark, and the site's commentary; the site accepted it (PROVED, last edited 7 April 2026). The partial result before it, Theorem 3 of Erdős, Sárközy and Sós (1989), which gives for at most three colors infinitely many squares that are sums of two distinct integers of one color, has the claim page Erdős, Sárközy and Sós 1989. The claim page Khalfalah and Szemerédi 2006 records the theorem, its postings and its acceptance evidence, the refereed publication and the documented acceptance, and the frontmatter standing derives from it.