Prime numbers are already unusual. They become less frequent as numbers grow, resist simple patterns, and continue to generate questions that look elementary but remain unresolved.
Yet hidden among the primes is a far rarer species, the Wilson primes.
Only three are known:
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What makes them special can be stated surprisingly simply.
For every prime p, Wilson’s theorem guarantees that (p − 1)! + 1 is divisible by p:
But Wilson primes satisfy a rarer condition: the same number is divisible by p²:
Exciting article! Anyone frustratingly grinding away at the mathematical search for the 4th Wilson prime might want to shift perspective and approach the problem from a purely geometric angle.
From a purely analytical standpoint, the classical…
I adore your statistical approach. Another interesting question is: what distribution (the family of distributions, plus the parameters) would nicely account for 5, 13, and 563? What does the 95% confidence interval of that distribution is the largest p one would need to search to find the next Wilson prime?
