In the above 128 x 128 matrix the n-th square is black iff the
number 10^50 + 2n + 1 is prime. By the prime number theorem,
the density of primes in this range is approximatively 1/log(10^50)
= 1/115, so we would expect about 2^15/115 = 285 primes
in this picture. The actual number is 269. The first prime after
10^50 is 10^50 + 151.
There are three twin primes (10^50 + x_i, 10^50 + x_i + 2) in this picture, with x_i = 18307, 19891, 29749. This is in good agreement with a conjecture of Hardy and Littlewood, which gives the density of twin primes as 1.32/log(x)^2. The expected number of twin primes in our interval of length 2^15 = 32768 according to this conjecture is 3.26.. .
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