🔢 Prime Distribution and Gaps

About this simulation

Every positive integer up to N is rendered as a small cube. Composite numbers stay dim and grey; primes light up in blue. As N grows, the primes visibly thin out — exactly as the Prime Number Theorem predicts, with density near 1/ln(N). Meanwhile the bars on the right track how often each even gap size between consecutive primes has occurred so far, building a live histogram.

How it works

• A sieve of Eratosthenes computes primality for every integer up to the slider's maximum.
• Wrapped number line lays integers out row by row, left to right, like a number line folded into a grid.
• Ulam spiral winds integers outward from the centre; primes tend to cluster along certain diagonals, a pattern Stanisław Ulam noticed doodling in 1963.
• Gaps between consecutive primes (2, 4, 6, 8…) are tallied live into the side histogram — small gaps dominate, but rare large gaps do occur.
• Twin primes (pairs differing by exactly 2, like 17 and 19) can be highlighted in gold.

Controls

• Upper limit N — how far along the integers the scene renders (30–3000).
• Scan speed — how fast N increases per second when auto-scan is on.
• Layout — switch between the grid number line and the Ulam spiral.
• Highlight twin primes — colours twin-prime pairs gold instead of blue.
• Auto-scan N — animates N growing continuously so you can watch the histogram fill in real time.

Did you know?

Even though primes become rarer on average, the largest known prime gaps grow far slower than the numbers themselves — and whether infinitely many twin primes exist remains one of the oldest unsolved problems in mathematics.