  For a finite set of non-zero natural numbers that contains at least one
element different from 1 and the least common multiple of any of its subsets,
there exists a subset of at least half of its members which has a common
divisor larger than 1. Utilizing a representation of the natural numbers as an
order-theoretical ring of prime power sets, this conjecture is shown to be
equivalent to Frankl's union-closed sets conjecture. Some results for cases
where the conjecture, which also has meaningful interpretations in graph and
lattice theory, is known to hold are provided. An equivalent dual version of
the conjecture is, that for a finite set of non-zero natural numbers that
contains at least two elements and the greatest common divisor of any of its
subsets, one of its members has a prime power that is not a prime power of more
than half of the members.
