[Math Lair] Twin Primes

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Twin primes are a special kind of prime numbers. They are pairs of primes of the form P and P + 2. There are several examples in the realm of small numbers; for example, 3 and 5, 5 and 7, 11 and 13, 17 and 19, 29 and 31, etc. A somewhat larger pair is 91,811 and 91,813.

Twin primes pose several questions that are still unanswered. For example, it is unknown whether there are an infinite number of twin primes. We do know that, as the natural numbers increase, the percentage of primes that are twin primes decreases, but it is not known whether twin primes thin out until they disappear entirely. The largest known twin primes are 697053813 × 2 16352 +/- 1, which have 4932 digits each. They were discovered in 1994.

We know that the harmonic series (the sum of the reciprocals of all natural numbers) diverges, which is to say that it approaches infinity as more and more terms are added. As well, the sum of the reciprocals of all prime numbers also diverges. What about the series formed by the sum of the reciprocals of all twin primes? The twin prime series is definitely sparser than the series using all the primes. Even though we don't know whether this series contains an infinite number of terms or not, we do know that it converges. In 1919, V. Brun proved that it converged to a value that has been calculated to approximately 1.90216. Another thing that is unknown is whether this number is rational, irrational (but algebraic) or transcendental. Brun was also able to prove that for any large number, somewhere in the sequence of natural numbers there is a succession of that many primes that are not twin primes.