Problem 4

Given a set

M

1985

distinct positive integers, none of which has a prime divisor greater than

23

, prove that

M

contains four distinct elements whose product is the fourth power of an integer.

Step 3 of 4: Classify the square roots again

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In plain words

The same compression is applied to the square roots. More roots than parity boxes force two roots to fit the same pattern.

s_j=t_j^2,\qquad t_j\mapsto(\text{prime exponents of }t_j)\pmod2

Detailed analysis

For each square-product pair write $x_jy_j=s_j^2$ . Classify the square root $s_j$ by the parity vector of its prime exponents. There are again only $2^9$ classes, but we have $2^9+1$ pair roots, so two, say $s_p,s_q$ , have the same class. Thus $s_ps_q$ is a square, say $u^2$ .