This is a backfill note from 2019-07-28. The idea is, what are mathematical achievements that are useful for motivating the study of mathematics?

Motivating problems in math

Solved and important!

• Infinite number of primes
• Sqrt(2) is irrational
• Fundamental theorem of algebra (1806)
• Unsolvability of the quintic (1824)
• Three construction problems of antiquity (1837, 1837, 1882)
• e and pi are transcendental (1873, 1882)
• Public-key cryptography (Diffie-Hellman-Merkle, RSA, elliptic curves) (1973-1976)
• Fermat’s Last Theorem / modularity theorem (1994, 2001)
• Poincare Conjecture (2003)
• Classification of finite simple groups (2004)

Axiomatics

• Parallel Postulate (1829-1868)
• Gödel’s Incompleteness Theorems (1931)
• Continuum Hypothesis (1940, 1963)

Solved and unimportant, but with an interesting anecdote

• Hilbert’s 3rd Problem (Dehn Invariants, 1883-1900, first to be solved)
• Four color theorem (1976)
• Kepler conjecture and honeycomb conjecture (1998, 1999, Thomas Hales). (Update 2026-07-03: sphere packing in dimension 8 actually earned Maryna Viazovka the 2022 Fields Medal, due to connections the work has with quasimodular forms. See the original paper and coverage in Plus and Quanta.)

Unsolved

• Goldbach Conjecture
• Twin prime conjecture
• Perfect numbers / Mersenne primes
• Collatz Conjecture
• Whether Euler-Mascheroni constant is rational / algebraic or not
• Riemann Hypothesis
• Ramsey numbers (R(5, 5), R(6, 6))
• ABC Conjecture
• Birch and Swinnerton-Dyer Conjecture

Update 2026-07-03: Added note on sphere packing in dimension 8.