Twin primes: classical results and new developments

While the notion of prime numbers has existed for millennia, twin primes have only been around for little over a century. Although it is not known whether there are infinitely many twin primes, the prime gap was very recently shown to be no greater than 246. The fact that the summed reciprocals of twin primes converge to approximately 1.9 has also been demonstrated. It has further been established that there do exist infinitely many primes p for which p+2 is the product of no more than two primes. A criterion for twin primes does exist but it is neither sufficient to show the existence of an infinite number of them, nor feasible as a computational tool.