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Statement

Setting (p. 2). Equation (2.1) asks for

$$\prod_{i=1}^{j}\prod_{l=0}^{k_i-1}(x_i+l)=y^2$$

in positive integers $x_1,\dots,x_j$ subject to condition (2.2): $x_s<x_t$ implies $x_s+k_s\le x_t$, so that blocks with distinct starting points are disjoint. The setting is stated on the Theorem 2.1 page.

Theorem 2.2 (p. 2). "If $j \geq 3$ and $k_i = 4$ for $1 \leq i \leq j$ then equation (2.1) has infinitely many solutions with (2.2)."

The paper says this confirms a conjecture of Ulas, who had proved the statement for $j=4$ and $j\ge6$ (Enseign. Math. 51 (2005), 331--334). The authors state as a belief, without proof, that their families for $j=3$, $(k_1,k_2,k_3)=(4,4,4)$ are minimal in $j$ among counterexamples to the Erdős--Graham proposal (p. 2), and in Section 5 (p. 6) they guess that for $j=2$ and $k_1\ge4$ equation (2.1) has at most finitely many solutions with (2.2). They also remark (p. 5) that their $j=3$ examples with $\max\{x_i\}<X$ number at most $c\log X$ for some constant $c$, whereas Ulas's $j=4$ examples number more than $X^\theta$ for some $\theta>0$.

Proof pointer

Section 4, pp. 4--5. Given the solution $x_1=33$, $x_2=1680$ for $j=2$, $k_1=k_2=4$, and Ulas's cases, it suffices to treat $j=3$. The paper gives three explicit families: two from solutions of $u^2-3v^2=-2$ (equation (4.1), with $u\equiv1$ or $u\equiv-1 \pmod 4$ and $u$ large enough for (2.2)) and one from odd solutions of $u^2-5v^2=4$ (equation (4.2)), expressed through Lucas and Fibonacci numbers. In each family the product of the three blocks is an explicit square of a polynomial in $u$ with rational coefficients.

Read depth

Claims checked: the setting, the hypotheses and the statement were read clause by clause on the page images of the print, and the proof was followed for structure. The polynomial identities were not recomputed. Nothing here is independently reviewed.

Dependencies

Ulas, On products of disjoint blocks of consecutive integers, Enseign. Math. 51 (2005), 331--334, for $j=4$ and $j\ge6$; not in the corpus.

Source. M. Bauer and M. A. Bennett, On a question of Erdős and Graham, Enseign. Math. (2) 53 (2007), 259--264; the edition read, paged 1--6, is named on the source card.

Bears on

• Problem 363: the problem asks whether only finitely many collections of disjoint intervals of at least four integers have a square product. Theorem 2.2 gives, for each fixed $j\ge3$, infinitely many collections of $j$ disjoint intervals of exactly four integers whose product is a square.