Show that there exists a set
![A](/media/m/5/a/e/5ae81275ee67d638485e903bdc0e9cde.png)
of positive integers with the following property: for any infinite set
![S](/media/m/c/6/3/c63593c3ec0773fa38c2659e08119a75.png)
of primes, there exist two positive integers
![m](/media/m/1/3/6/1361d4850444c055a8a322281f279b39.png)
in
![A](/media/m/5/a/e/5ae81275ee67d638485e903bdc0e9cde.png)
and
![n](/media/m/a/e/5/ae594d7d1e46f4b979494cf8a815232b.png)
not in
![A](/media/m/5/a/e/5ae81275ee67d638485e903bdc0e9cde.png)
, each of which is a product of
![k](/media/m/f/1/3/f135be660b73381aa6bec048f0f79afc.png)
distinct elements of
![S](/media/m/c/6/3/c63593c3ec0773fa38c2659e08119a75.png)
for some
![k \geq 2](/media/m/c/8/0/c804b0e182a8747bb8bb61bf115bf9f2.png)
.
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Show that there exists a set $A$ of positive integers with the following property: for any infinite set $S$ of primes, there exist two positive integers $m$ in $A$ and $n$ not in $A$, each of which is a product of $k$ distinct elements of $S$ for some $k \geq 2$.