Can the product of 4 distinct numbers be equal to a fourth power?

Question

The set $M$ consists of 2001 distinct positive integers, none of which is divisible by any prime $p > 23$. Prove that there are distinct $x, y, z, t \in M$ such that $xyzt = u^4$ for some integer $u$.

I guess some kinda Pigeonhole Principle argument should be useful, but I can't figure it out.

Answer 1

Not an answer, but an approach that I believe will work. As you say in the comments, each of the numbers in $M$ corresponds to a nine position vector with each position being the powers of $2,3,5,\ldots 23$ in the factorization of the number reduced $\bmod 4$. We are looking for four vectors that sum to the zero vector when each entry is reduced $\bmod 4$.

Let us concentrate on the first position in the vector. There are $2001 \choose 4$ ways to choose a four element subset of $M$. If the numbers in $M$ are distributed among the four residues, the sums will be (almost) equidistributed among $0,1,2,3$, so $\frac 14$ of the sums will be zero. Looking at all the entries in the vector we expect ${2001 \choose 4}\frac 1{4^9} \gt 2\ 500\ 000$ sums to be zero in all entries.

To avoid the equidistribution we can have all the elements of $M$ in two classes. If half have entry $0$ and half have entry $1$ in a position, the only way to get a sum of zero is to select all four with the same entry. This is only $\frac 18$ of the sets of four but you still get ${2001 \choose 4}\frac 1{8^9}\approx 4962$ sums to the zero vector.

To make this solid, let there be $a$ numbers with first entry $0, b$ with first entry $1, c$ with first entry $2,$ and $d$ with first entry $3$. You have $a+b+c+d=2001$. To get a sum of $0$ you can select four of the same, giving ${a \choose 4}+{b \choose 4}+{c \choose 4}+{d \choose 4}$ sets, or two with residue $1$ and two with residue $3$ giving ${b \choose 2}{d \choose 2}$ sets, or a number of other possibilities. You need to show that at least $\frac 1{20}$ of the sets sum to zero because $20^9 \lt {2001 \choose 4}$

Answer 2

Just think of even/odd.

There are 512 classes of integers, depending on the parities of the nine exponents. So two must belong to the same class. Pair them off, you have 1999 numbers left.
Another two belong to the same class as each other because 1999>512. And so on until only 511 numbers are left.

You now have $(2001-511)/2$ pairs of numbers, the product of each pair is a square.

Now, how to show that two of the pairs have a product which is a fourth power...