2c:[] 2a:{"title":"Gilbreath Conjecture","cells":[{"id":0,"value":"# Gilbreath Conjecture","pinned":false,"hidden":false,"mode":"md"},{"id":43,"value":"The Gilbreath conjecture, also known as the Proth–Gilbreath conjecture, is an unsolved conjecture in number theory about the sequence of prime numbers. It states that if we repeatedly take the absolute differences between consecutive primes, the first value in every resulting row is always 1.","pinned":false,"hidden":false,"mode":"md"},{"id":78,"value":"{\n const n = 8;\n\n let row = [];\n for (let i = 2; row.length < n; i++) if (isPrime(i)) row.push(i);\n\n let rows = [];\n while (row.length) {\n rows.push(row.join(\" & \"));\n row = row.slice(1).map((v, i) => Math.abs(v - row[i]));\n }\n\n return tex`\n \n \\begin{aligned}\n \\begin{matrix}\n ${rows.join(String.raw`\\\\`)}\n \\end{matrix}\n \\end{aligned}\n\n `;\n}","pinned":false,"hidden":false,"mode":"ojs"},{"id":10,"value":"{\n const z = 128;\n return Pixel8.canvas(\n (px8) => {\n const primes = [];\n for (let n = 2; primes.length < z; n++) {\n if (isPrime(n)) primes.push(n);\n }\n let cr = primes;\n for (let y = 0; y < px8.h; y++) {\n for (let x = 0; x < cr.length; x++) {\n px8.pset(x, y, cr[x]);\n }\n const nr = [];\n for (let i = 0; i < cr.length - 1; i++)\n nr.push(Math.abs(cr[i + 1] - cr[i]));\n cr = nr;\n }\n },\n { w: z, h: z, s: 3, paused: true }\n );\n}","pinned":false,"hidden":false,"mode":"ojs"},{"id":49,"value":"---\n### Thomas Matrix\n\nUnlike the Gilbreath difference triangle, which is built by repeatedly taking differences of the previous row, a related construction known as the Thomas matrix was introduced by James Thomas in 2004. It grows from boundary values: the first row is initialized with the prime numbers, the first column is fixed at 2, and every remaining cell is computed as the absolute difference between its left neighbor and its upper-left neighbor. From these simple local rules, Sierpinski triangle shows up.\n\nFurther reading: [Prime Puzzles - Sierpinski triangles from prime numbers](https://www.primepuzzles.net/puzzles/puzz_274.htm).\n","pinned":false,"hidden":false,"mode":"md"},{"id":13,"value":"{\n const z = 128;\n return Pixel8.canvas(\n (px8) => {\n let prev = new Array(z).fill(2);\n for (let c = 0; c < z; c++) px8.pset(0, c, prev[c]);\n let n = 3;\n for (let r = 1; r < z; r++) {\n while (!isPrime(n)) n++;\n const cur = new Array(z);\n cur[0] = n;\n n++;\n for (let c = 1; c < z; c++) {\n cur[c] = Math.abs(cur[c - 1] - prev[c - 1]);\n }\n for (let c = 0; c < z; c++) {\n px8.pset(r, c, cur[c]);\n }\n prev = cur;\n }\n },\n { w: z, h: z, s: 3, paused: true }\n );\n}","pinned":false,"hidden":false,"mode":"ojs"},{"id":56,"value":"---","pinned":false,"hidden":false,"mode":"md"},{"id":16,"value":"function isPrime(n) {\n if (n <= 1) return false;\n if (n <= 3) return true;\n if (!(n & 1) || n % 3 === 0) return false;\n for (let i = 5; i * i <= n; i += 6) {\n if (n % i === 0 || n % (i + 2) === 0) return false;\n }\n return true;\n}","pinned":false,"hidden":false,"mode":"ojs"},{"id":7,"value":"import { Pixel8 } from \"@nxrix/pixel-8\"","pinned":false,"hidden":false,"mode":"ojs"}],"files":"$W2c","stdlib":"1","titling":"auto"}