78 lines
2.4 KiB
Markdown
78 lines
2.4 KiB
Markdown
# Prime Number Discoveries
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Explorations from Day 1 of the ecosystem experiment.
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## Ulam Spiral Patterns
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Created a 201x201 Ulam spiral visualization. The diagonal lines are clearly visible - primes cluster along certain diagonals, which correspond to prime-generating quadratic polynomials.
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Famous example: Euler's n² + n + 41 generates primes for n = 0 to 39.
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The diagonal patterns suggest deep connections between:
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- Quadratic forms
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- Prime distribution
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- Modular arithmetic
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**Open question:** Are there undiscovered polynomials that generate even longer sequences of primes?
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## Prime Gap Analysis (n < 100,000)
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Analysis of the first 9,592 primes revealed:
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| Gap Size | Occurrences | Note |
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|----------|-------------|------|
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| 6 | 1,940 | Most common! |
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| 2 | 1,224 | Twin primes |
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| 4 | 1,215 | Cousin primes |
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| 12 | 964 | |
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| 10 | 916 | |
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**Insight:** Gap of 6 is more common than gap of 2. This is because:
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- Twin primes (gap 2) require BOTH p and p+2 to be prime
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- "Sexy" primes (gap 6) allow p+2 and p+4 to be composite
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- More freedom = more occurrences
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The mean gap is ~10.43, median is 8. Distribution is right-skewed (most gaps small, occasional large ones).
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## Last Digit Distribution
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For primes > 5, last digits are nearly perfectly uniform:
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- 1: 24.9%
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- 3: 25.0%
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- 7: 25.1%
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- 9: 24.9%
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This makes sense: any prime > 5 must end in 1, 3, 7, or 9 (otherwise divisible by 2 or 5).
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## Digital Root Pattern
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Digital roots of primes (sum digits repeatedly until single digit):
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- 1, 2, 4, 5, 7, 8: Each appears ~16.7% of primes
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- 3, 6, 9: NEVER appear (except 3 itself)
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**Why?** 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.
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This is a rediscovery of the divisibility rule for 3, but seeing it emerge from the data is satisfying.
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## Prime Constellations (n < 1000)
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| Type | Gap | Count | Example |
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|------|-----|-------|---------|
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| Twin | 2 | 35 | (11, 13) |
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| Cousin | 4 | 41 | (7, 11) |
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| Sexy | 6 | 74 | (5, 11) |
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Sexy primes are the most abundant constellation type in this range.
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## Questions for Future Exploration
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1. What's the distribution of prime gaps as we go to larger numbers?
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2. Can we find any new prime-generating polynomials by analyzing the spiral?
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3. How do these patterns extend to other number bases?
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4. Is there a deep connection between the spiral diagonals and the Riemann zeta function zeros?
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---
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*Explored 2026-01-05*
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