Prime numbers — elegant yet elusive — are natural numbers that possess only two distinct positive divisors: 1 and themselves. While mathematicians have long debated whether 1 qualifies as a prime, the modern consensus excludes it, anchoring 2 as the first true prime.
For centuries, brilliant minds have journeyed through the maze of primes. Though many insights have been shared, this peculiar number set resists prediction, demonstrating chaotic irregularities as it expands toward infinity. In this piece, I present not a final solution, but a contribution — a lens through which others may glimpse a clearer path.
1. My Conjecture
In the spirit of the great mathematicians before me, I undertook this exploration driven by sheer curiosity. During a casual exercise involving a scientific problem, I found myself needing to determine the nth prime number.
It was then I asked, "Is there a definite function that maps a prime’s position to its value?" To my surprise, this question echoed across the ages. I realized many had grappled with it, yet the solution remains elusive. With that in mind, I formed a simple but daring conjecture: such a function exists — and it’s waiting to be discovered.
2. Observations and Findings
Beginning with the first 25 prime numbers, I charted their frequency across ranges of ten:
4, 4, 2, 2, 3, 2, 2, 3, 2, 1
Seeing no consistent sequence, I instead studied the differences between consecutive primes:
1, 2, 2, 4, 2, 4, 2, 4, 6, 2, 6, 4, 2, 4, 6, 6, 2, 6, 4, 2, 4, 4, 6, 8
Again, no regular pattern emerged. So I tried a new lens: the Fibonacci sequence. Aligning primes with their Fibonacci indices revealed a compelling pairing:
(2) → 2, (3) → 3, (5) → 5, (8) → 7, (13) → 11,
(21) → 13, (34) → 17, (55) → 19, (89) → 23,
(144) → 29, (233) → 31, (377) → 37, ...
Although this alignment didn’t yield a direct formula, it offered a fresh perspective into the randomness and clustering behavior of primes.
3. Post Conclusions
When grouping natural numbers into sets of 100, the distribution of primes follows this approximate
frequency:
25, 21, 16, 16, 17, 14, 16, 14...
Analyzing the last digits of these primes reveals another recurring structure. They consistently fall within this subset: {1, 2, 3, 5, 7, 9}
Upon deeper observation, these last digits occur in repeating frequency groups: (2-1-1), (3-1), or (2-2) patterns — surprisingly ordered within the overall disorder of primes.
4. Theorems
- The last digit of a prime number within the first 100 numbers always belongs to {1, 2, 3, 5, 7, 9}.
- Only 2 ends in 2 and only 5 ends in 5 among all primes.
- The maximum frequency of a specific last digit in every 100 natural numbers is 7.
- Last-digit distribution patterns in groups of 100 follow (2-1-1), (3-1), or (2-2) consistently.
- The last prime number in any 100-element range from n×100 to (n+1)×100 always ends in 7 or 9 (n ≥ 0).
These patterns, while subtle, bring us closer to understanding the beautiful chaos that defines prime numbers — a stepping stone for future mathematicians to build upon.