Brun's theorem

Brun's theorem states that the sum of the reciprocals of all twin primes converges to a finite value, now called Brun's constant. It does not establish whether infinitely many twin primes exist.

Twin prime: A pair of prime numbers that differ by two. These are the prime pairs whose reciprocals Brun's theorem sums.
Viggo Brun: A Norwegian mathematician who developed influential sieve methods in number theory. Brun proved the theorem that bears his name.
Twin prime conjecture: The conjecture that infinitely many prime pairs differ by two. Brun's theorem leaves this central existence question unanswered.
Harmonic series: The divergent series formed by summing the reciprocals of the positive integers. Its divergence contrasts with the convergent reciprocal sum over twin primes.
Brun's sieve: A sieve method that gives upper bounds for integers satisfying several primality-like conditions. Brun used sieve estimates to bound the number of twin-prime pairs.
Prime number theorem: A theorem describing the asymptotic distribution of primes among the positive integers. It provides the wider historical problem of quantifying prime distribution.
Reciprocal primes: The reciprocals of prime numbers, whose sum over all primes diverges. Their divergent sum contrasts with convergence over the much sparser twin-prime pairs.
Euler's theorem on the infinitude of primes: Euclid's theorem proving that infinitely many primes exist. Infinitude of primes is known, unlike infinitude of twin primes.
Sieve theory: A branch of number theory that estimates sets of integers by excluding residue classes associated with divisibility. Brun's proof belongs to the broader family of sieve methods.
Hardy–Littlewood conjectures: Conjectures predicting asymptotic formulas for prime patterns, including twin primes. Their predicted twin-prime density is much sharper than Brun's bound.

Linked from 2 pages