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 Math Help > Number Theory > Prime > Reciprocals of Primes > Small Set

A "large" subset of positive integers is one whose sum-of-reciprocals diverges, whereas a "small" subset of positive integers has a convergent sum-of-reciprocals.  The odd positive integers, for example, is a "large" set.  The squares, on the other hand, is "small".

What about the numbers, base r, that can be written without using one particular digit, d?  Is this set large or small?  Answer: the set is small -- the set of reciprocals of these numbers converges.

The sum of reciprocals of numbers that can be written without using digit d, base r, converges.

Let S1 be the sum of reciprocals of one-digit numbers, base r, that can be written without using digit d.

Let S2 be the sum of reciprocals of two-digit numbers, base r, that can be written without using digit d.

etc.

S1 is the sum of r-2 reciprocals, none bigger than 1, so S1 ≤ r-2

S2 is the sum of (r-2)(r-1) reciprocals, none bigger than 1/r, so S2 ≤ (r-2)(r-1/r)

S3 is the sum of (r-2)(r-1)2 reciprocals, none bigger than 1/r2, so S3 ≤ (r-2)(r-1/r)2

etc.

So S1 + S2 + S3 + ... ≤ (r-2)(  1 + (r-1/r) + (r-1/r)2 + ... ) = (r-2)(r)

Facts about Small and Large sets

A union of any finite number of small sets is a small set.

A subset of a small set is a small set.

The compliment of a small set in a large set is large.  That is, if L is a large set, and S is a small set, then the set L-S is large, where L-S consists of all elements of L that are not in S.

Any large set can be partitioned into two (or any finite number of) large sets, all of which are large.  (One method is to assign elements of L in order to L1, L2, ..., Ln, L1, L2, ..., Ln, in "round robin" fashion).

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