Riemann Hypothesis, round n^9^9^9!
The Riemann hypothesis, one of the millennium prizes offered by the Clay Mathematics Institute, has had a long and varied history attempts upon it: the conjectures really boil down to assessing how close the current approximation of n / ln n primes less than a given number the solution truly is. So it ends up being one of those open-ended p/non-p categories of problems. However, some of the attempts exhibit interesting cultural implications, especially when filtering in the pressure to celebritize.
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