Math Explorations

Mathematical Principles & Theorems

Explores famous unsolved and computational mathematical phenomena: (1) The Collatz Conjecture (\(3n+1\)): \(T(n) = n/2\) if even, \(3n+1\) if odd, hypothesizing all positive integers reach the cycle \(4 \to 2 \to 1\); (2) Ulam Prime Spiral: arranging integers on a square spiral reveals dense quadratic polynomial prime concentrations (Euler's polynomial \(n^2+n+41\)); (3) Riemann Zeta function partial sums \(\zeta(s) = \sum_{n=1}^N n^{-s}\).

Operating Instructions

• Select exploration module: Collatz Hailstone Trajectories, Ulam Prime Spiral, or Riemann Zeta.
• Configure seed value \(N\) or partial sum term count using numeric inputs.
• For Collatz: View maximum altitude, trajectory step count, and sequence line charts.
• For Ulam Spiral: Inspect prime diagonal alignments and compute prime density metrics.