Every finite subset of the positive integers is small.
The set of all positive integers is a large set; this statement is equivalent to the divergence of the
harmonic series. More generally, any
arithmetic progression (i.e., a set of all integers of the form an + b with a ≥ 1, b ≥ 1 and n = 0, 1, 2, 3, ...) is a large set.
The set of
square numbers is small (see
Basel problem). So is the set of
cube numbers, the set of 4th powers, and so on. More generally, the set of positive integer values of any
polynomial of degree 2 or larger forms a small set.
The set {1, 2, 4, 8, ...} of powers of
2 is a small set, and so is any
geometric progression (i.e., a set of numbers of the form of the form abn with a ≥ 1, b ≥ 2 and n = 0, 1, 2, 3, ...).
The set of
prime powers which are not prime (i.e., all numbers of the form pn with n ≥ 2 and p prime) is small although the primes are large. This property is frequently used in
analytic number theory. More generally, the set of
perfect powers is small; even the set of
powerful numbers is small.
The set of numbers whose expansions in a given
base exclude a given digit is small. For example, the set
of integers whose
decimal expansion does not include the digit 7 is small. Such series are called
Kempner series.
The
union of finitely many small sets is small, because the sum of two
convergent series is a convergent series. (In set theoretic terminology, the small sets form an
ideal.)
Paul Erdősconjectured that all large sets contain arbitrarily long
arithmetic progressions. He offered a prize of $3000 for a proof, more than for any of his
other conjectures, and joked that this prize offer violated the minimum wage law.[1] The question is still open.
It is not known how to identify whether a given set is large or small in general. As a result, there are many sets which are not known to be either large or small.
Every finite subset of the positive integers is small.
The set of all positive integers is a large set; this statement is equivalent to the divergence of the
harmonic series. More generally, any
arithmetic progression (i.e., a set of all integers of the form an + b with a ≥ 1, b ≥ 1 and n = 0, 1, 2, 3, ...) is a large set.
The set of
square numbers is small (see
Basel problem). So is the set of
cube numbers, the set of 4th powers, and so on. More generally, the set of positive integer values of any
polynomial of degree 2 or larger forms a small set.
The set {1, 2, 4, 8, ...} of powers of
2 is a small set, and so is any
geometric progression (i.e., a set of numbers of the form of the form abn with a ≥ 1, b ≥ 2 and n = 0, 1, 2, 3, ...).
The set of
prime powers which are not prime (i.e., all numbers of the form pn with n ≥ 2 and p prime) is small although the primes are large. This property is frequently used in
analytic number theory. More generally, the set of
perfect powers is small; even the set of
powerful numbers is small.
The set of numbers whose expansions in a given
base exclude a given digit is small. For example, the set
of integers whose
decimal expansion does not include the digit 7 is small. Such series are called
Kempner series.
The
union of finitely many small sets is small, because the sum of two
convergent series is a convergent series. (In set theoretic terminology, the small sets form an
ideal.)
Paul Erdősconjectured that all large sets contain arbitrarily long
arithmetic progressions. He offered a prize of $3000 for a proof, more than for any of his
other conjectures, and joked that this prize offer violated the minimum wage law.[1] The question is still open.
It is not known how to identify whether a given set is large or small in general. As a result, there are many sets which are not known to be either large or small.