Squares from blocks of consecutive integers: a problem of Erdős and Graham

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lemma_2_1: Bennett and Van Luijk's lemma that, for a positive squarefree integer D < 100, the equation 2(4t^2+t-4) = Ds^2 has infinitely many solutions in positive integers s and t exactly when D is one of 5, 7, 13, 37, 47, 58, 67, 73, 83 and 97.

theorem_1_1: Bennett and Van Luijk's theorem that for every r >= 5 there are infinitely many tuples (n_1, ..., n_r, x) of positive integers with f(n_1,5)...f(n_r,5) = x^2 and n_j + 5 <= n_{j+1}, where f(n,k) is the product of the k consecutive integers starting at n.

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Bennett, Michael A. and Van Luijk, Ronald, Squares from blocks of consecutive integers: a problem of Erdős and Graham. Indag. Math. (N.S.) 23 (2012), 123--127. The copy read for this card is the authors' six-page manuscript, paged 1--6 rather than with the journal's page numbers; it prints no notice. The publisher's page for the journal version could not be read on 2026-10-02 (DOI 10.1016/j.indag.2011.11.002), and its Crossref record lists only Elsevier's text-and-data-mining and open-archive user licenses, which govern the version of record, not this file; the term is unstated. Labels and pages on this card and its result pages are the manuscript's.

With $f(n,k)=n(n+1)\cdots(n+k-1)$, Erdős and Graham asked whether equation (1), $\prod_{j=1}^r f(n_j,k_j)=x^2$, has, for fixed $r\geq1$ and fixed $k_1,\ldots,k_r$ with every $k_j\geq4$, at most finitely many solutions in positive integers $(n_1,\ldots,n_r,x)$ with $n_j+k_j\leq n_{j+1}$ for $1\leq j\leq r-1$, condition (2) (p. 1). Theorem 1.1 (p. 2) answers this in the negative for blocks of length five: if $r\geq5$ and $k_i=5$ for all $i$, there are infinitely many $(r+1)$-tuples of positive integers $(n_1,\ldots,n_r,x)$ satisfying (1) and (2).

The authors say it is unclear whether the polynomial-identity and Pell-equation argument that Bauer and Bennett used for blocks of length four extends to blocks of length five or more (p. 2). They instead look for four polynomials $p_i(t)\in\mathbb Z[t]$ with $p_i(t)\neq p_j(t)+k$ for $1\leq i,j,k\leq4$, $i\neq j$, whose product of five-term blocks is $g(t)h(t)^2$ with $\deg g\leq2$ (equation (4), p. 2), and find one family with $g(t)=2(4t^2+t-4)$ (p. 4). Since $g(t)$ is never a square modulo $25$, they solve $g(t)=Ds^2$ instead (Lemma 2.1, p. 4: for positive squarefree $D<100$ there are infinitely many positive solutions exactly for ten values of $D$), attach fixed blocks with product $Dy^2$ for $r\in\{5,6,7,9,10,11,12\}$, and induct from $r-8$ (p. 5). The authors note that Ulas suggested infinitely many solutions whenever $r$ exceeds a constant depending only on $\max_i k_i$ (pp. 1--2), that they know of no solution with all $k_i=5$ and $r\leq3$ while $r=4$ has many, and that their techniques appear unlikely to work for blocks of length six or more (pp. 5--6).

Source: https://personal.math.ubc.ca/~bennett/publ.html.

Read status: claims checked for Theorem 1.1 and Lemma 2.1, read clause by clause on the page images of the manuscript; the proof of Theorem 1.1 was followed and rests on Lemma 2.1, whose proof the paper only sketches. A search made for this corpus found solutions of Lemma 2.1's equation for $D=5,13,58,97$ only with $t$ negative, against the lemma's "positive integers $s$ and $t$"; such $t$ still give positive tuples for Theorem 1.1 (see its page). Nothing here is independently reviewed.

Bears on. #363: Theorem 1.1 (p. 2) gives, for each fixed $r\geq5$, infinitely many collections of $r$ disjoint blocks of five consecutive positive integers with square product, which the paper presents as a negative answer to Erdős and Graham's finiteness question for blocks of five.

Results.

• Theorem 1.1 (p. 2): for $r\geq5$ and all $k_i=5$, there are infinitely many $(r+1)$-tuples of positive integers satisfying (1) and (2).
• Lemma 2.1 (p. 4): for positive squarefree $D<100$, the equation $2(4t^2+t-4)=Ds^2$ has infinitely many solutions in positive integers exactly when $D\in\{5,7,13,37,47,58,67,73,83,97\}$.

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