Problem 363

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claims/: The 4 claim pages of Problem 363, one per claimant's result; the problem's standing derives from them.

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Statement. Is it true that there are only finitely many collections of disjoint intervals $I_1,\ldots,I_n$ of size $\lvert I_i\rvert \geq 4$ for $1\leq i\leq n$ such that

$$\prod_{1\leq i\leq n}\prod_{m\in I_i}m$$

is a square?

Formulation. The site's wording leaves the number $n$ of intervals and their sizes free. The standing answers that wording, which is false. Skałba (Colloq. Math. 98 (2003), 1--3, Theorem 2) shows that once the number of blocks may vary, disjoint blocks of any fixed length $l\geq 4$ have a square product infinitely often (claim page).

Erdős and Graham's question, as Bauer and Bennett and Bennett and Van Luijk state it and as the formal-conjectures statement file encodes it, fixes $n$ and the sizes $k_1,\dots,k_n\geq 4$. It asks whether each such choice admits only finitely many collections. The answer is also no. Ulas's blocks of four for $n=4$ and $n\geq 6$, Bauer and Bennett's for $n=3$ and $n=5$, and Bennett and Van Luijk's blocks of five for $n\geq 5$ each give infinitely many collections for one fixed choice.

Status. DISPROVED (LEAN). The "(LEAN)" suffix is the site's catalog label; the disproofs, their acceptance and the Lean formalization are recorded on the claim pages, and the corpus has built no Lean for this problem.

Source. erdosproblems.com/363, accessed 2026-09-04. Cite as: T. F. Bloom, Erdős Problem #363, https://www.erdosproblems.com/363.

References.

• [BaBe07] Bauer, Mark and Bennett, Michael A., On a question of Erdős and Graham. Enseign. Math. (2) (2007), 259-264.
• [BeVL12] Bennett, Michael A. and Van Luijk, Ronald, Squares from blocks of consecutive integers: a problem of Erdős and Graham. Indag. Math. (N.S.) (2012), 123-127.
• [Ul05] Ulas, Maciej, On products of disjoint blocks of consecutive integers. Enseign. Math. (2) (2005), 331-334.

Formalization. Statement in formal-conjectures, pinned to the repository's revision of 2026-10-06.

Current assessment

The question (site formulation of 2026-09-04). The statement above; DISPROVED (LEAN), page last edited 2 December 2025. The question is Erdős and Graham's. The site's commentary recalls, as context, the theorem of Erdős and Selfridge that a product of two or more consecutive integers is never a perfect power, and explains why blocks of at least four are required: Pomerance observed that the four blocks of three consecutive integers, two centered at $2^{n-1}$ and $2^n$ and two ending at $2^{2n-1}$ and $2^{2n}$, always have square product.

Claims. Four refereed papers each give infinitely many collections and so each disproves the statement as worded: Skałba (2003), with the number of blocks free; Ulas (2005), for $n=4$ and every $n\geq 6$ blocks of four, the disproof the site's curator credits; Bauer and Bennett (2007), for $n=3$ and $n=5$ blocks of four, so for every $n\geq 3$; and Bennett and Van Luijk (2012), for every $n\geq 5$ blocks of five. The last three also answer the fixed-$n$ reading. Each of the last three pages lists refereed evidence and the curator's credit as reviewed; the site does not credit Skałba, so his page lists refereed alone. Ulas conjectured that, for every fixed block size, the solutions become infinite in number once $n$ is large enough; that conjecture is open, and Problem 930 asks a more general question.

Formalization and the Lean label. The site's thread carries a Lean 4 file posted on 2026-03-10 by Wouter van Doorn, obtained with the system Aristotle from Harmonic, proving that one of Ulas's parametrizations gives an infinite family; the formal-conjectures statement file, which fixes the number and sizes of the intervals, marks erdos_363 solved and names a copy in Boris Alexeev's repository, which declares itself a formalization of Ulas's result, as its formal proof. Ulas's claim page records both. Their Lean theorem says the set of valid collections is infinite, with the number of intervals free. Its validity predicate does not exclude an interval containing $0$, whose product $0$ is a square, so the statement alone is weaker than either reading. The substance is the proof that Ulas's family consists of valid collections of positive integers. The corpus has not built or audited either file.

Search scope. 2026-10-07: the site's problem page, commentary and discussion thread, with the Crossref record of [BeVL12], the e-periodica volume records of [Ul05] and [BaBe07], the publisher's record of Skałba's paper, and the repositories' commit records for dates. arXiv, MathSciNet, zbMATH, Google Scholar and X were not searched.

Remaining gaps. (1) The corpus holds no copy of Ulas's or Skałba's paper, and their claim pages cite the journals' records; the statements of the other two papers are on their library cards. (2) The corpus has not built the Lean files, so no page lists formalized evidence. (3) Ulas's general conjecture, blocks of any fixed size for large $n$, is open beyond sizes four and five.

Linked library material

These entries are derived from explicit links on library pages. They are navigation only and do not by themselves record mathematical progress.

• bauer_2007_question_erdos_graham
• bauer_2007_question_erdos_graham / theorem_2_1
• bauer_2007_question_erdos_graham / theorem_2_2
• bennett_2012_squares_blocks_consecutive_integers_problem_erdos
• bennett_2012_squares_blocks_consecutive_integers_problem_erdos / lemma_2_1
• bennett_2012_squares_blocks_consecutive_integers_problem_erdos / theorem_1_1
• erdos_1976_products_factorials
• erdos_1976_products_factorials / question_p337