Here we list some proofs (or just the proven theorems; or even just a mathematical topic) which seem to be worth to be studied in connection with Hilbert’s 24th Problem. Please, feel free to suggest further example (just write an e-mail to: kahle@fct.unl.pt).

• Syzygies and Syzygies of Syzygies.
  Hilbert’s own suggestion.
• Infinity of Prime Numbers.
  Classical example from Euclid via Euler to Fürstenberg which posseses quite different proofs; also discussed in the context of purity.
• Fundamental Theorem of Algebra.
  Gauß alone provided four different proofs.
• Euler’s Theorem.
  Its standard proof is surprising simplicity, yet not just an instance of induction.
• Cantor-Bernstein Theorem
• …

The classical book of Aigner and Ziegler, Proofs from THE BOOK, may contain more examples.