Eight unsolved mathematical problems

Eight things to think about…

1) Are there there infinitely many twin prime pairs p and p+2? We have 3/5, 5/7, 11/13, 17/19, etc. Does that sequence go on forever?

2) Are there there infinitely many Sophie Germain prime pairs p and 2p+1? We have 2/5, 3/7, 5/11, 11/23, etc. Does that sequence go on forever?

3) Is every even number greater than 4 the sum of two odd primes? We have 6 = 3 + 3, 8 = 3 + 5, 10 = 3 + 7, 12 = 5 + 7, etc. Does that always work?

4) Are there infinitely many perfect numbers? We have 6 = 1 + 2 + 3, 28 = 1 + 2 + 4 + 7 + 14, 496, 8128, 33550336, etc. Does that sequence go on forever? And are there any odd numbers in the sequence?

5) Is π a normal number? That is, do the digits 0 to 9 occur equally often in 3.14159265358979323846264338327950288419716939937510582…, and does that also work for bases other than decimal?

6) Do the nontrivial zeros of the Riemann zeta function all have real part ½?

7) Does P = NP? This is perhaps the most important (and most famous) unsolved problem in computer science, and there is a million-dollar prize for solving it.

8) Do smooth solutions to the Navier–Stokes equations always exist? There is a million-dollar prize for this one too.

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