|
The Copeland–Erdős constant is the concatenation of "0." with the base 10 representations of the prime numbers in order. Its value is approximately :0.235711131719232931374143… . The constant is irrational; this can be proven with Dirichlet's theorem on arithmetic progressions or Bertrand's postulate (Hardy and Wright, p. 113) or Ramare's theorem that every even integer is a sum of at most six primes. It also follows directly from its normality (see below). By a similar argument, any constant created by concatenating "0." with all primes in an arithmetic progression ''dn'' + ''a'', where ''a'' is coprime to ''d'' and to 10, will be irrational. E.g. primes of the form 4''n'' + 1 or 8''n'' + 1. By Dirichlet's theorem, the arithmetic progression ''dn''·10''m'' + ''a'' contains primes for all ''m'', and those primes are also in ''cd'' + ''a'', so the concatenated primes contain arbitrarily long sequences of the digit zero. In base 10, the constant is a normal number, a fact proven by Arthur Herbert Copeland and Paul Erdős in 1946 (hence the name of the constant). The constant is given by : where ''pn'' is the ''n''th prime number. Its continued fraction is (4, 4, 8, 16, 18, 5, 1, … ) (). ==Related constants== In any given base ''b'' the number : which can be written in base ''b'' as 0.0110101000101000101…''b'' where the ''n''th digit is 1 if ''n'' is prime, is irrational. (Hardy and Wright, p. 112). 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「Copeland–Erdős constant」の詳細全文を読む スポンサード リンク
|