JavaScript is off, so the live parts of this page are not running. Everything written below is complete on its own and says what each computation does and what it produced — the empty rectangles are where it would have run in front of you. Nothing here needs an account, and nothing is hidden behind the scripts; they compute, they do not reveal.

Moving visuals on this page — measured at 0.26% brightness change per frame and zero flashes per second; your reduce motion setting stops them. What moves, and how it was measured

Mathematical plates that draw themselves — prime counts, orbits and lattices rendered live rather than shipped as pictures. It runs continuously while on screen, and it pauses on its own whenever it scrolls out of view.

We measured it rather than guessing: the largest frame-to-frame change in average brightness is 0.26%, and it produces zero flashes per second against the threshold of three set by WCAG 2.3.1. That is well under the limit, and it is not a promise about any individual: if moving images are a problem for you, you should not have to assess ours.

Everything animated on this site honours your system’s reduce motion setting — turn it on in your operating system or browser and these pages stop moving, with nothing else lost. All the mathematics is in the text and the still figures.

561, the smallest Carmichael number: 2^35 mod 561 squared 4 times gives 263, 166, 67, 1, 1. The non-trivial square root of 1 is what proves it composite.

Miller–Rabin visualizer Numbers & structure

Watch a primality test reason about a number, step by step

A different Miller–Rabin than the RSA demo above — that one picks 20 random witnesses each run; this one always tests the same 12 smallest primes (2..37), so every witness and every intermediate squaring can be shown and reasoned about, not just asserted. That fixed base list is proven (Sorenson & Webster, 2015) to give zero false positives — not a probability — for any number below 3.3×10²⁴; only above that (this tool accepts up to 400 digits) does it become a strong empirical heuristic rather than a proven bound. The verdict below says which regime your number is in. Type a number and press Test primality.

Math: every witness round below shows the actual $n-1 = d\cdot 2^{r}$ split ( $d$ odd) the whole test hinges on, then squares $a^{d} \bmod n$ up to $r-1$ times looking for a nontrivial square root of 1 — the one thing an odd prime can never have. Drag the witness slider down and you're watching the exact same computation with fewer independent checks, not a different algorithm. Dev: modular exponentiation by repeated squaring, BigInt throughout, same discipline as the exact calculator above.

witnesses to test: 12 your own witness a= (optional, tested first)

miller-rabin — verdict

witness trace

raw output, click to copy

← Previous toolCollatz trajectory explorer

Back to the tools · The Notebook, if you would rather read than compute.

Name	Type	Set by	Purpose	Duration
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`ksec_tool_uses_…`	localStorage	This site	A per-tool daily-use counter for the free tools, one key per tool (a fair-use nudge, not an enforced limit, never sent anywhere) — set only if you actually use one of those tools	Resets daily; until you clear it
`__cf_bm` and similar	HTTP cookie	Cloudflare, Inc. (our security & CDN provider)	Distinguishes legitimate traffic from automated abuse — the same security gate described in Methodology	Session, typically under 30 minutes