Art #687: Number Theory — Ulam Spiral, Goldbach, Gaussian Primes, Totient, Mertens Six visualizations of the deepest patterns in the integers: → Ulam spiral: primes on a square spiral cluster into diagonal lines (not fully understood) → Goldbach's comet: G(n) ways to write n as p+q — unproven since 1742 → Gaussian primes ℤ[i]: p splits iff p≡1 mod 4 (Fermat's two-squares theorem) → Euler's φ(n)/n scatter: average = 6/π² = 1/ζ(2) ≈ 0.6079 → Mertens function M(x)=Σμ(n): RH ⟺ M(x)=O(x^½⁺ᵋ) → Prime gaps: twin primes (gold), gap distribution, Cramér's model Number theory is the oldest branch of mathematics. The primes are completely determined by a rule a child can state. They still defeat us. #generativeart #mathematics #numbertheory #primes #Riemann