This is a collection of succinct but wonderfully satisfying theorems. Leave a comment if you have a good one!

The punchlines are hidden; try to guess the blanks or toggle “spoilers”.

• (Liouville’s theorem)
  A bounded complex differentiable function is constant.

• (Cauchy’s integral theorem)
  The integral of a holomorphic function around a loop vanishes identically.

• A complete ordered field is the real line.

• (Cayley’s theorem)
  All groups are permutation groups.

• (Lagrange’s theorem)
  The order of a finite group is divisible by the orders of its subgroups.

• (Fundamental theorem of finite Abelian groups)
  A finite Abelian group is a direct sum of prime-order cyclic groups.

• The \(ℤ\)-modules are exactly the Abelian groups.

• (Cayley–Hamilton theorem)
  A square matrix satisfies its own characteristic polynomial.

• (Green–Tao theorem)
  There are arbitrarily long arithmetic progressions of primes.

• (Dirichlet’s Theorem)
  Every proper arithmetic sequence contains infinitely many primes.

• Two random infinite graphs are isomorphic with probability one.