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Prime numbers are the prime example (no pun intended) for something that arises apparently without describable patters; we know that infinitely many exist, that gaps between them can be arbitrarily large, but also that no matter how far you go, you will always eventually find small gaps.

What are other examples of objects in mathematics that crop up without any discernible patters, but are totally unconnected to prime numbers?

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    $\begingroup$ The stable homotopy groups of the sphere are always a good example of an important mathematical structure with no evident patters (of course this doesn't mean there aren't deeper structures behind running the show) $\endgroup$
    – Denis Nardin
    Commented Dec 22, 2019 at 8:37
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    $\begingroup$ There are plenty of discernible patterns in the primes, of which you even mentioned some, and the prime number theorem provides more. So the current question is ill-posed. $\endgroup$
    – user44143
    Commented Dec 22, 2019 at 8:39
  • $\begingroup$ Possibly ranks of elliptic curves? $\endgroup$
    – LeechLattice
    Commented Dec 22, 2019 at 9:01
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    $\begingroup$ You mean like digits of $\pi$? $\endgroup$
    – Gerald Edgar
    Commented Dec 22, 2019 at 14:21
  • $\begingroup$ I want to say, "non-trivial zeros of the Riemann zeta-function," but the part about "totally unconnected to prime numbers" stops me dead in my tracks. $\endgroup$
    – Gerry Myerson
    Commented Dec 22, 2019 at 15:53

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Finite simple groups.$\mbox{}\mbox{}$

(Not totally unrelated to prime numbers but more general...)

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