Why the Ulam spiral has diagonals
The prime spiral on the home page, explained
Write the integers in a square spiral and the primes line up along diagonals. Two short arguments explain most of the pattern, and an open conjecture covers the rest.
antony-mathematics
Antony
Number theory, analysis, and whatever I'm proving this week.
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All posts →The Basel problem, and where the π comes from
Why the sum of 1/n² equals π²/6
A proof of the Basel sum using Fourier series and Parseval's identity, plus an interactive look at how slowly the partial sums converge.
Three proofs that there are infinitely many primes
Euclid's, Euler's, and one that uses topology
The same theorem proved three ways: by multiplication, by divergence of the harmonic series, and by a topology on the integers.
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From the notebook
All notes →- Nov 2, 2026 Odd numbers make squares
- Oct 30, 2026 Wilson’s theorem says (p-1)! \equiv -1 \pmod p for every prime p. The slickest proof I know: in \F_p^\times every element pairs off with its inverse, except the two elements…
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Prove that the fraction \dfrac{21n + 4}{14n + 3} is irreducible for every natural number n.
Choose any n + 1 numbers from \{1, 2, \dots, 2n\}. Prove that one of the chosen numbers divides another.
Two diagonally opposite corner squares are cut from an 8 \times 8 chessboard, leaving 62 squares. Can the rest be covered exactly by 31 dominoes, each covering two adjacent squares?
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