Exercise 4.5.10* (There is a consecutive block of integers in the sequence whose product is a perfect square)

Proof. Let $a_1,a_2,\dots ,a_n$ be any sequence of positive integers. Let $k$ be the total number of distinct prime factors of the product of the integers. Then we can write

Let ${\alpha }_{i,j}=\overline{u_{i,j}}$ denote the class of $u_{i,j}$ in the field ${𝔽}_2=\mathbb{Z}∕2\mathbb{Z}$. Then the product $\underset{j=i}{\overset{l}{\prod <!--\; FUNCTION\; APPLICATION\; -->}}{a}_j$ of a consecutive block of integers in the sequence is a perfect square if and only if

Let us write $w_i=({\alpha }_{i,1},{\alpha }_{i,2},\dots ,{\alpha }_{i,k})\in {𝔽}_2^k$ for $1\le i\le n$. Then the previous condition is written more concisely in the form

Consider now the cumulative sums

(In particular, $s_0=\underset{i\in \varnothing }{\sum <!--\; FUNCTION\; APPLICATION\; -->}{w}_i=0$). There are $n+1$ such sums $s_0,s_1,\dots ,s_n$ in the vector space ${𝔽}_2^k$, which has $2^k$ elements. By hypothesis, $n+1>2^k$. Therefore, the map

cannot be injective (one-to-one) so there are two sums

with the same values (this is the “pigeonhole principle”). Then

This shows that $a_{i+1}{a}_{i+2}\cdots a_j$ is a perfect square. If $n\ge 2^k$, there is a consecutive block of integers in the sequence $(a_1,a_2,\dots ,a_n)$ whose product is a perfect square. □