In 1984, Pat Costello [2] also showed how to construct Smith numbers in
the form n = P Q 10k, where P is a small prime and Q is a Mersenne
prime. The idea behind this construction is quite simple: If PQ is not
already Smith, keep multiplying PQ by 10. Doing so will not alter S(PQ),
while it repeatedly adds 7 to Sp(PQ)|hopefully until we reach the equality
S(n) = Sp(n), or else we try again with a dierent P. In fact, the role of
a Mersenne prime here is not really needed|unless one aims for a record
prime to generate a largest Smith number.
In 1984, Pat Costello [2] also showed how to construct Smith numbers inthe form n = P Q 10k, where P is a small prime and Q is a Mersenneprime. The idea behind this construction is quite simple: If PQ is notalready Smith, keep multiplying PQ by 10. Doing so will not alter S(PQ),while it repeatedly adds 7 to Sp(PQ)|hopefully until we reach the equalityS(n) = Sp(n), or else we try again with a di erent P. In fact, the role ofa Mersenne prime here is not really needed|unless one aims for a recordprime to generate a largest Smith number.
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