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There's also the pattern that every prime (above 3) modulo 6 is congruent to 1 or 5. I suspect that some of the pattern comes from that fact.


Is there a more generalized version of that pattern, perhaps something like this (I haven't checked):

  let P(n) be the nth prime number: P(1)=2, etc.
  let n be an integer >= 2, and m be an integer > n
  then P(m) ≡ 1 and (P(n) - 1) (mod P(n))

?




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