The Collatz Conjecture (3n+1 problem) asks: take any positive integer — if even, halve it; if odd, triple it and add 1. Repeat. Does every starting number eventually reach 1? Despite its simplicity, no proof exists. This tree maps the reverse process, showing how all integers trace back to 1 through ×2 and (n−1)÷3 paths. Primes glow as diamonds; branch color reflects depth from the trunk. Right-click any node to expand the tree beyond the initial range.

Floyd's triangle arranges consecutive natural numbers in rows of increasing length — row 1 has one number, row 2 has two, and so on. Named after Robert Floyd, the pattern reveals how primes distribute across a triangular grid. Each column follows a quadratic sequence, and prime clustering varies intriguingly by row and column position. Watch as the triangle builds row by row, primes glowing gold among dim blue composites.

In 1963, mathematician Stanisław Ulam arranged consecutive integers in a spiral and highlighted the primes — revealing unexpected diagonal patterns. These diagonals correspond to quadratic polynomials that generate unusually many primes, a phenomenon that remains unexplained. Watch as the spiral builds outward layer by layer, primes glowing gold against dim blue composites.

An Ulam spiral, but the integers are not placed in natural order. Instead, they are laid out by their depth in the reverse Collatz tree — layer by layer, starting from 1, with each layer's new numbers sorted ascending. Because the Collatz tree is believed to reach every positive integer, this gives an alternative permutation of ℕ that covers the same numbers as the classic Ulam spiral. Do the famous prime diagonals survive, break, or form something new? Drag the slider to grow the tree deeper.

Primes are arranged on an Ulam spiral grid, but instead of highlighting which numbers are prime, each cell displays the gap to the next prime — how many composites lie between consecutive primes. Twin primes (gap 1) glow blue while the largest gaps burn red, with a 10-color spectrum revealing the full landscape of prime spacing. The result maps prime density into a color field where patterns in the gaps become visible at a glance.

In 1742, Christian Goldbach conjectured that every even integer greater than 2 can be written as the sum of two primes. For each even number, we count how many such prime pairs exist — 4=2+2 has just one, while 100=3+97=11+89=17+83=... has many. Plotting these counts produces a striking comet shape with distinct density bands, revealing hidden structure in how primes combine. Nearly three centuries later, the conjecture remains unproven.