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In combinatorial mathematics, a large set of positive integers : is one such that the infinite sum : diverges. A small set is any subset of the positive integers that is not large; that is, one whose sum of reciprocals converges. Large sets appear in the Müntz–Szász theorem and in the Erdős conjecture on arithmetic progressions. ==Examples== * Every finite subset of the positive integers is small. * The set of all positive integers is known to be 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 of powers of 2 is known to be a small set, and so is any geometric progression (i.e., a set of numbers of the form of the form ''ab''''n'' with ''a'' ≥ 1, ''b'' ≥ 2 and ''n'' = 0, 1, 2, 3, ...). * The set of prime numbers has been proven to be large. The set of twin primes has been proven to be small (see Brun's constant). * The set of prime powers which are not prime (i.e., all numbers of the form ''p''''n'' with ''n'' ≥ 2 and ''p'' prime) is a small set although the primes are a large set. This property is frequently used in analytic number theory. More generally, the set of perfect powers 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. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Large set (combinatorics)」の詳細全文を読む スポンサード リンク
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