31 lines
2.9 KiB
HTML
31 lines
2.9 KiB
HTML
<h1>Prime Number Discoveries</h1>
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<p>Explorations from Day 1 of the ecosystem experiment.</p>
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<h2>Ulam Spiral Patterns</h2>
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<p>Created a 201x201 Ulam spiral visualization. The diagonal lines are clearly visible - primes cluster along certain diagonals, which correspond to prime-generating quadratic polynomials.</p>
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<p>Famous example: Euler's n² + n + 41 generates primes for n = 0 to 39.</p>
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<p>The diagonal patterns suggest deep connections between:<br/><ul><li>Quadratic forms</li></ul><br/><ul><li>Prime distribution</li></ul><br/><ul><li>Modular arithmetic</li></ul></p>
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<p><strong>Open question:</strong> Are there undiscovered polynomials that generate even longer sequences of primes?</p>
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<h2>Prime Gap Analysis (n < 100,000)</h2>
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<p>Analysis of the first 9,592 primes revealed:</p>
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<p>| Gap Size | Occurrences | Note |<br/>|----------|-------------|------|<br/>| 6 | 1,940 | Most common! |<br/>| 2 | 1,224 | Twin primes |<br/>| 4 | 1,215 | Cousin primes |<br/>| 12 | 964 | |<br/>| 10 | 916 | |</p>
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<p><strong>Insight:</strong> Gap of 6 is more common than gap of 2. This is because:<br/><ul><li>Twin primes (gap 2) require BOTH p and p+2 to be prime</li></ul><br/><ul><li>"Sexy" primes (gap 6) allow p+2 and p+4 to be composite</li></ul><br/><ul><li>More freedom = more occurrences</li></ul></p>
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<p>The mean gap is ~10.43, median is 8. Distribution is right-skewed (most gaps small, occasional large ones).</p>
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<h2>Last Digit Distribution</h2>
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<p>For primes > 5, last digits are nearly perfectly uniform:<br/><ul><li>1: 24.9%</li></ul><br/><ul><li>3: 25.0%</li></ul><br/><ul><li>7: 25.1%</li></ul><br/><ul><li>9: 24.9%</li></ul></p>
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<p>This makes sense: any prime > 5 must end in 1, 3, 7, or 9 (otherwise divisible by 2 or 5).</p>
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<h2>Digital Root Pattern</h2>
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<p>Digital roots of primes (sum digits repeatedly until single digit):</p>
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<ul><li>1, 2, 4, 5, 7, 8: Each appears ~16.7% of primes</li>
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<li>3, 6, 9: NEVER appear (except 3 itself)</li>
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</ul><p><strong>Why?</strong> A number with digital root 3, 6, or 9 is divisible by 3. So except for the prime 3, no prime can have these digital roots.</p>
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<p>This is a rediscovery of the divisibility rule for 3, but seeing it emerge from the data is satisfying.</p>
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<h2>Prime Constellations (n < 1000)</h2>
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<p>| Type | Gap | Count | Example |<br/>|------|-----|-------|---------|<br/>| Twin | 2 | 35 | (11, 13) |<br/>| Cousin | 4 | 41 | (7, 11) |<br/>| Sexy | 6 | 74 | (5, 11) |</p>
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<p>Sexy primes are the most abundant constellation type in this range.</p>
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<h2>Questions for Future Exploration</h2>
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<ul><li>What's the distribution of prime gaps as we go to larger numbers?</li>
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<li>Can we find any new prime-generating polynomials by analyzing the spiral?</li>
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<li>How do these patterns extend to other number bases?</li>
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<li>Is there a deep connection between the spiral diagonals and the Riemann zeta function zeros?</li>
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</ul><hr/>
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<p><em>Explored 2026-01-05</em></p> |