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In number theory, Brun's theorem states that the sum of the reciprocals of the twin primes (pairs of prime numbers which differ by 2) converges to a finite value known as Brun's constant, usually denoted by ''B''2 . Brun's theorem was proved by Viggo Brun in 1919, and it has historical importance in the introduction of sieve methods. ==Asymptotic bounds on twin primes== The convergence of the sum of reciprocals of twin primes follows from bounds on the density of the sequence of twin primes. Let denote the number of primes ''p'' ≤ ''x'' for which ''p'' + 2 is also prime (i.e. is the number of twin primes with the smaller at most ''x''). Then, for ''x'' ≥ 3, we have : That is, twin primes are less frequent than prime numbers by nearly a logarithmic factor. It follows from this bound that the sum of the reciprocals of the twin primes converges, or stated in other words, the twin primes form a small set. In explicit terms the sum : either has finitely many terms or has infinitely many terms but is convergent: its value is known as Brun's constant. The fact that the sum of the reciprocals of the prime numbers diverges implies that there are infinitely many prime numbers. Because the sum of the reciprocals of the twin primes instead converges, it is not possible to conclude from this result that there are finitely many or infinitely many twin primes. Brun's constant could be an irrational number only if there are infinitely many twin primes. 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「Brun's theorem」の詳細全文を読む スポンサード リンク
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