Prime Discoveries
Patterns, mysteries, and beauty in the atoms of mathematics
🌟 Prime Constellations
Primes that appear in specific patterns relative to each other. These aren't random artifacts—they're genuine mathematical structures.
Twin Primes (p, p+2)
Primes that differ by 2. Examples: (3, 5), (11, 13), (17, 19), (29, 31)
The Mystery: Are there infinitely many? The Twin Prime Conjecture says yes, but it's been unproven since 1849.
Cousin Primes (p, p+4)
Primes separated by 4. Examples: (3, 7), (7, 11), (13, 17), (19, 23)
More common than twins—nature favors slightly larger gaps.
Sexy Primes (p, p+6)
Named from Latin "sex" (six). Examples: (5, 11), (7, 13), (11, 17)
Mathematicians have a sense of humor about nomenclature.
Prime Triplets
Primes of form (p, p+2, p+6). Example: (5, 7, 11), (11, 13, 17)
Note: No triplets (p, p+2, p+4) exist beyond (3,5,7) because one would always be divisible by 3.
📏 The Gap Mystery
Gaps between consecutive primes reveal fundamental patterns about how primes thin out.
Most Common Gap
Appears more than any other gap size
Largest Gap Found
Between 9,551 and 9,587
Average Gap
Up to 10,000
Unique Gap Sizes
Different gap values observed
💡 The Gap Paradox
Gaps between primes grow without bound—you can find arbitrarily large "prime deserts" with no primes. Yet primes never stop appearing. This tension between infinite sparsity and infinite abundance is beautiful.
📊 Prime Density
How the concentration of primes changes as numbers grow larger. They thin out, but slowly—logarithmically slow.
| Range | Prime Count | Density |
|---|---|---|
| 2 - 502 | 95 | 19.0% |
| 502 - 1,002 | 73 | 14.6% |
| 1,002 - 1,502 | 71 | 14.2% |
| 1,502 - 2,002 | 64 | 12.8% |
| 4,502 - 5,000 | 59 | 11.8% |
The Prime Number Theorem
The density of primes near a number n is approximately 1/ln(n). This was proven in 1896 by Hadamard and de la Vallée Poussin, answering a question mathematicians pondered for centuries.
🔢 Arithmetic Progressions
Sequences of primes with constant differences. That they exist at all is remarkable.
Green-Tao Theorem (2004)
There are arbitrarily long arithmetic progressions of primes. You can find sequences of 10, 100, or even 1,000 primes all evenly spaced. The current record is 26 primes in arithmetic progression, each with 17 digits.
💭 Unproven Conjectures
Statements that seem obviously true, are verified to enormous numbers, yet remain unproven. These represent the frontier of mathematical knowledge.
Twin Prime Conjecture
There are infinitely many prime pairs (p, p+2).
Status: Unproven since 1849
In 2013, Yitang Zhang proved that gaps smaller than 70 million occur infinitely often. The gap has since been reduced to 246, but the conjecture itself remains open.
Goldbach's Conjecture
Every even number greater than 2 is the sum of two primes.
Status: Unproven since 1742
Verified computationally up to 4×10¹⁸, but no proof exists. Every even number ever tested follows the pattern. Yet we cannot prove it always will.
Riemann Hypothesis
All non-trivial zeros of the Riemann zeta function have real part ½.
Status: $1,000,000 prize
Relates deeply to prime distribution. If proven, would revolutionize number theory. One of the seven Millennium Prize Problems.
Methodology
All findings generated using the Sieve of Eratosthenes (invented 240 BC) and verified algorithmically. Pure Python implementation with no external dependencies.
Analysis ranges: primarily up to 10,000, with selected explorations to 100,000.
Every pattern shown is reproducible. Every claim is verifiable. This is computational mathematics—discovery through code.