There’s something about this special sub-set in the integers set known as prime numbers, unyielding and yet endlessly mysterious. It feels deeply personal. As if the integers are whispering ancient secrets through a language only a few are tuned to understand. Where others have seen chaos, I see some scattered patterns beckoning us to listen more differently and closely. This is my journey, not a conclusion, but a conjecture, a set of observations, a map for others and me to carry further into the unknown.

1. My Conjecture: The Pulse Beneath the Primes

I began with a simple, persistent question: “How can I find the nth prime number?” At first glance, this might appear naïve. It is a question that has echoed through centuries, from the classrooms of ancient Alexandria to the chalkboards of modern-day mathematicians. But in that moment, I wasn’t standing on the shoulders of giants, I was staring into the void, trying to feel the pulse beneath the primes.

I believe there exists a deterministic function, a hidden mapping, that can link the position of a prime number in the sequence (its index in ℙ, the prime set) directly to its value. A closed-form expression. Not probabilistic, not approximate. Exact.

What I propose is not definitive. It is not final. But it is a step, a conjecture, that invites scrutiny, refinement, and maybe, just maybe, discovery.

2. Observations and Findings: Listening to the Noise

I started with raw inspection. The first 25 prime numbers were arranged, not for beauty, but for behaviour. I split them into segments, decades of the number line, and counted how many primes occurred in each:

Distribution across decadal intervals:
[2-10]: 4 primes | [11-20]: 4 | [21-30]: 2 | [31-40]: 2 | [41-50]: 3 | [51-60]: 2 | [61-70]: 2 | [71-80]: 3 | [81-90]: 2 | [91-100]: 1

A curious oscillation. No obvious order, but certainly no chaos either.

Then I traced the gaps between successive primes, the prime difference sequence:
1, 2, 2, 4, 2, 4, 2, 4, 6, 2, 6, 4, 2, 4, 6, 6, 2, 6, 4, 2, 4, 4, 6, 8

A symphony of twos and fours, punctuated by sixes and the occasional eight. It was not random, it was a structured irregularity. It called to mind quantum fluctuations—predictable in form, uncertain in detail.

Then came a bold idea: what if primes and the Fibonacci sequence intersected more deeply than we imagined?

I aligned the Fibonacci sequence with the prime index positions:

Fibonacci (n):   2, 3, 5, 8, 13, 21, 34, 55, 89, ...
Prime at (Fn):   3, 5, 7, 11, 13, 17, 19, 23, 29, ...

There, too, was a poetic rhythm, Fibonacci growth echoing in the sequence of primes. Was it mere coincidence or an indication of a deeper combinatorial resonance?

3. Post-Conclusions: A Pattern Beyond Patterns

Taking a macroscopic view, I examined the count of primes in successive blocks of 100 natural numbers:
[1–100]: 25 primes | [101–200]: 21 | [201–300]: 16 | [301–400]: 16 | [401–500]: 17 | [501–600]: 14 ...

A decaying sequence, but not monotonically. It resembles an exponential decay model with local fluctuations, suggesting entropy in a deterministic system.

Even more intriguing were the terminal digits of primes. Except for 2 and 5 (the only even prime and the only prime divisible by 5), primes always ended in:
1, 3, 7, or 9

Why not 0, 4, 6, or 8? Simple divisibility constraints eliminate them. But the frequency of occurrence among (1, 3, 7, 9) appeared to follow repeating clusters:
(2-1-1), (3-1), or (2-2)

This suggested modular symmetry, not just randomness, but the possibility of periodic distributions in base 10 representations.

4. Theorems and Hypotheses

Let me offer the following hypotheses, born of my analysis:

Theorem 1:
In any set of 100 consecutive natural numbers, all terminal digits of primes (except for 2 and 5) must be from the set {1, 3, 7, 9}.

Theorem 2:
The digits 2 and 5 only appear once each in the terminal position of prime numbers—specifically, as the numbers 2 and 5.

Theorem 3:
In a block of 100 numbers, no single terminal digit from the prime set {1, 3, 7, 9} occurs more than 7 times.

Theorem 4:
When 2 and 5 are excluded, the sequence of last-digit distributions in primes often follows recognisable micro-patterns such as (2-1-1), (3-1), or (2-2).

Conclusion: A Symphony of the Unsolved

Prime numbers are not just a curiosity; they are the bedrock of mathematical understanding, cryptography, randomness, and even the philosophy of certainty. Euclid gave us their infinitude. Gauss gave us their density. We shall one day unlock their soul.

This publication might not contain any solution, but the next stepping-stone to get across the river

This is my call to the next explorer: hear the music in the noise. Take these fragments, the way I took the fragments of Euclid, Euler, and Riemann, and go further. Primes do not reveal themselves easily—but to those who persist, they sometimes whisper.

Checkout my weekly blogs on mysteries of Math and Science in my blog Raymond's Thinking Palace