Prime Spirals

Ulam’s doodle and Euler’s diagonal

An interactive lesson with 7 steps, about 6 minutes: you predict, drag and solve, with instant feedback on every answer. It runs in your browser and needs JavaScript.

What you’ll learn

• Ulam’s spiral shows primes clustering along diagonals.
• Each diagonal follows a quadratic; some are rich in primes, others empty.
• $n^2 + n + 41$ is prime for $n = 0$ to $39$, but no polynomial is always prime.

Steps

1. A doodle in a dull meeting
2. Half the diagonals
3. Formulas on diagonals
4. Euler’s diagonal
5. The streak must end
6. No prime machine
7. Primes in polar coordinates