One day during my lunch break I wrote a simple Eratosthenes' sieve to count twin primes. For instance, the number of pairs up to 10¹³ is 15834664872. The sum of inverses of twin primes up to that limit is about 1.81394376067 which extrapolates to an approximation for Brun's constant: 1.90216055 or 6 or 7 or thereabouts. Here is a 4.5K gif of a graph showing how the extrapolated sum converges as the limit is increased. Since then, Thomas R. Nicely has published the results of similar computations up to 10^14...
Talking about sums-of-inverses-of-primes, you may know that the sum of 1/p for primes p <= x is close to B + log log x. As far as I know, nobody has previously computed B to any significant precision. however B =
I gave a short talk about frequent gaps between primes at the 1994 Western Number Theory Conference in San Diego. Read all about it here!
All that stuff is like five years old.
Now here are some interesting numbers: 92221507219705345685350, 276856274258963891889538, 333373190151749761757285479, 41871609686648820507900581, 112815344736959642274131295791, 37837308472231540269443981458, 64206457470187038759216338447, 47455661896223045299748316018941. Check 'em out! Anyway, they're discrete logarithms in the groups of points on some elliptic curves over some finite fields and they are the answers to challenges posed by Certicom...
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