30:T490,The Goldbach Conjecture is one of the oldest and most famous unsolved problems in mathematics. It is deceptively simple to state, yet it has resisted all attempts at a complete proof for nearly 300 years. First proposed by the German mathematician Christian Goldbach in a 1742 letter to Leonhard Euler, the conjecture actually comes in two versions. The **strong** (or binary) conjecture, which most people refer to today, states that every even integer greater than 2 can be expressed as the sum of two primes. The **weak** (or ternary) conjecture states that every odd integer greater than 5 can be written as the sum of three primes. The strong conjecture also implies the weak conjecture, so proving the strong version would automatically prove the weak version as well. In fact, if ${tex`(n > 5)`} is odd, then ${tex`(n-3)`} is even and greater than 2. If the strong conjecture is true, then ${tex`n - 3 = p + q`} for some primes (p) and (q). Therefore, ${tex`n = 3 + p + q`} which expresses (n) as the sum of three primes. Interestingly, the weak version was finally proven in 2013 by Harald Helfgott, but the strong version remains unproven to this day.31:T1517,{ const primes = nPrimes(9); const maxPrime = primes[primes.length - 1] + 2; const w = width - 48; const h = w * 0.75; const a = (Math.min(w, h) / (35 * maxPrime)) * 1.5; const margin = Math.min(w, h) * 0.125; const cx = w / 2; const cy = h / 2; const top = [cx, margin]; const right = [w - margin, cy]; const bottom = [cx, h - margin]; const left = [margin, cy]; const svg = d3 .create("svg") .attr("width", w) .attr("height", h) .attr("viewBox", `0 0 ${w} ${h}`); svg .append("polygon") .attr( "points", [top, right, bottom, left].map((p) => p.join(",")).join(" ") ) .attr("fill", "none") .attr("stroke", "currentColor") .attr("stroke-width", a * 2); function pointOnLine(a, b, t) { return [a[0] + (b[0] - a[0]) * t, a[1] + (b[1] - a[1]) * t]; } function intersection(a, b, c, d) { const e = (a[0] - b[0]) * (c[1] - d[1]) - (a[1] - b[1]) * (c[0] - d[0]); if (Math.abs(e) < 1e-10) return null; const x = ((a[0] * b[1] - a[1] * b[0]) * (c[0] - d[0]) - (a[0] - b[0]) * (c[0] * d[1] - c[1] * d[0])) / e; const y = ((a[0] * b[1] - a[1] * b[0]) * (c[1] - d[1]) - (a[1] - b[1]) * (c[0] * d[1] - c[1] * d[0])) / e; return [x, y]; } function inside(p) { const dx = Math.abs(p[0] - cx); const dy = Math.abs(p[1] - cy); const hw = right[0] - cx; const hh = cy - top[1]; return dx / hw + dy / hh <= 1 + 1e-9; } const linesA = []; const linesB = []; const sums = new Map(); primes.forEach((prime) => { const t = prime / maxPrime; const leftPoint = pointOnLine(top, left, t); const rightPoint = pointOnLine(top, right, t); const topPoint = pointOnLine(top, right, t); const bottomPoint = pointOnLine(left, bottom, t); svg .append("text") .attr("x", leftPoint[0] - a * 2) .attr("y", leftPoint[1] - a * 4) .attr("text-anchor", "end") .attr("font-size", a * 17) .attr("font-family", "var(--serif)") .attr("font-weight", "700") .attr("fill", "currentColor") .attr("stroke", "light-dark(#fff,#000)") .attr("stroke-width", a * 3) .attr("paint-order", "stroke") .attr("stroke-linejoin", "round") .text(prime); svg .append("text") .attr("x", rightPoint[0] + a * 2) .attr("y", rightPoint[1] - a * 4) .attr("text-anchor", "start") .attr("font-size", a * 17) .attr("font-family", "var(--serif)") .attr("font-weight", "700") .attr("fill", "currentColor") .attr("stroke", "light-dark(#fff,#000)") .attr("stroke-width", a * 3) .attr("paint-order", "stroke") .attr("stroke-linejoin", "round") .text(prime); svg .append("circle") .attr("cx", leftPoint[0]) .attr("cy", leftPoint[1]) .attr("r", a * 3) .attr("fill", "currentColor"); svg .append("circle") .attr("cx", rightPoint[0]) .attr("cy", rightPoint[1]) .attr("r", a * 3) .attr("fill", "currentColor"); const lineAStart = pointOnLine(top, left, t); const lineAEnd = pointOnLine(right, bottom, t); const lineBStart = pointOnLine(top, right, t); const lineBEnd = pointOnLine(left, bottom, t); linesA.push([lineAStart, lineAEnd]); linesB.push([lineBStart, lineBEnd]); svg .append("line") .attr("x1", lineAStart[0]) .attr("y1", lineAStart[1]) .attr("x2", lineAEnd[0]) .attr("y2", lineAEnd[1]) .attr("stroke", "currentColor") .attr("stroke-width", a * 1); svg .append("line") .attr("x1", lineBStart[0]) .attr("y1", lineBStart[1]) .attr("x2", lineBEnd[0]) .attr("y2", lineBEnd[1]) .attr("stroke", "currentColor") .attr("stroke-width", a * 1); }); for (let i = 0; i < primes.length; i++) { for (let j = 0; j < primes.length; j++) { const p = intersection( linesA[i][0], linesA[i][1], linesB[j][0], linesB[j][1] ); if (!p || !inside(p) || (i === 0) !== (j === 0)) { continue; } const sum = primes[i] + primes[j]; if (!sums.has(sum)) { sums.set(sum, { min: p[0], max: p[0], y: p[1] }); } else { const e = sums.get(sum); e.min = Math.min(e.min, p[0]); e.max = Math.max(e.max, p[0]); e.y = p[1]; } svg .append("text") .attr("x", p[0]) .attr("y", p[1] + 4) .attr("text-anchor", "middle") .attr("font-size", a * 17) .attr("font-family", "var(--serif)") .attr("font-weight", "700") .attr("fill", "currentColor") .attr("stroke", "light-dark(#fff,#000)") .attr("stroke-width", a * 3) .attr("paint-order", "stroke") .attr("stroke-linejoin", "round") .text(sum); } } /*function getVBoundaryX(y) { const hw = right[0] - cx; const hh = cy - top[1]; const dy = Math.abs(y - cy); if (dy > hh) { return null; } const dx = hw * (1 - dy / hh); return [cx - dx, cx + dx]; } for (const [sum, e] of sums) { svg .append("line") .attr("x1", e.min) .attr("y1", e.y) .attr("x2", e.max) .attr("y2", e.y) .attr("stroke", "currentColor") .attr("stroke-width", 2); }*/ return svg.node(); }32:T5f5,nPrimes = (n) => { if (n < 1) return []; if (n === 1) return [2]; const lnN = Math.log(n); const limit = Math.ceil(n * (lnN + Math.log(lnN))) + 10; const sqrtLimit = Math.floor(Math.sqrt(limit)); const baseSieveSize = (sqrtLimit - 1) >> 1; const baseSieve = new Uint8Array(baseSieveSize); const basePrimes = []; for (let i = 0; i < baseSieveSize; i++) { if (baseSieve[i] === 0) { const p = 2 * i + 3; basePrimes.push(p); const start = 2 * i * i + 6 * i + 3; for (let j = start; j < baseSieveSize; j += p) { baseSieve[j] = 1; } } } const primes = [2]; const segmentSize = 1 << 15; let low = 3; const sieve = new Uint8Array(segmentSize); while (low <= limit && primes.length < n) { const high = Math.min(low + segmentSize * 2 - 1, limit); const currentSieveSize = (high - low + 1) >> 1; sieve.fill(0, 0, currentSieveSize); for (let i = 0; i < basePrimes.length; i++) { const p = basePrimes[i]; let start = p * p; if (start < low) { let diff = low % p; start = diff === 0 ? low : low + (p - diff); if ((start & 1) === 0) start += p; } let startIdx = (start - low) >> 1; for (let j = startIdx; j < currentSieveSize; j += p) { sieve[j] = 1; } } for (let i = 0; i < currentSieveSize; i++) { if (sieve[i] === 0) { primes.push(low + 2 * i); if (primes.length === n) break; } } low += segmentSize * 2; } return primes; }33:[] 2e:{"title":"Goldbach's conjecture","cells":[{"id":0,"value":"# Goldbach's conjecture","pinned":false,"hidden":false,"mode":"md"},{"id":214,"value":"$30","pinned":false,"hidden":false,"mode":"md"},{"id":4,"value":"$31","pinned":false,"hidden":false,"mode":"ojs"},{"id":211,"value":"Prime pairs ${tex`(p,q)`} and their sums ${tex`(p+q)`}, visualizing the strong Goldbach conjecture.","pinned":false,"hidden":false,"mode":"md"},{"id":7,"value":"$32","pinned":false,"hidden":false,"mode":"ojs"}],"files":"$W33","stdlib":"1","titling":"auto"}