The Six-Lane Highway
Primes, Patterns, and Riemann — Frequency Waves in Prime Distributions
Introduction
Prime numbers generate striking patterns when we visualise the multiples of each prime as "frequency
waves" or "vibrating strings".
These waves are perfectly regular along the entire number line. When overlaid, they create zigzags,
pyramids, spirals, and tubular structures — revealing deep, non-random order in the distribution of
composites while primes remain as the untouched "gaps".
Early Frequency Waves Beginning With the Number 1 Frequency Wave
When lining up all odd numbers (prime and non-prime) in a six-lane formation, with each number the
start of a new wave, I noticed a zigzag pattern like a frequency wave, starting with prime number 3.
The finding of #3-frequency wave was followed by the #5-frequency, and displayed here several more.
They all originate from the zero point, and oscillate around a central "zero line".
(vertical spacings are added for easier viewing)
Pyramid Patterns
The Pyramid strip overlays the number 1 frequency wave. It determines exactly where,
and how far along the strip, the frequency insertion points and primes are located.
The #1 wave (odd numbers) overlaid by the others form pyramid structures consisting of predictable
peaks and base ratios which have a consistent rhythm of 2-4-2-4-
— curving toward infinity.
It effectively separates and sifts out the prime numbers from the non-prime numbers.
Examples:
Frequency #1 Odd numbers — one to infinity
Frequency #3 placed on the #1 frequency overlays non-prime numbers to 25 (5²)
Next place #5 frequency, it overlays non-prime numbers to 49 (7²)
Next place #7 frequency, it overlays non-prime numbers to 121 (11²)
Next place #11 frequency, it overlays non-prime numbers to 169 (13²)
Next place #13 frequency, it overlays non-prime numbers to 289 (17²)
Next place #17 frequency, it overlays non-prime numbers to 361 (19²)
Next place #19 frequency, it overlays non-prime numbers to 529 (23²)
Etc.
Imagine a train moving along a train track and passing known and predetermined stops along the way.
At each stop a new frequency is engaged to cover all non-prime numbers on the strip up to the next
stop and ahead of the train. As the distances between stops increase continually, the train can speed
up since there are fewer stops and frequencies required.
Placing progressively longer sections of track ahead of a faster and faster moving train, and disappearing
into some kind of event horizon, shows that the Riemann Hypothesis that posits that all non-trivial zeros
of the Zeta Function lie on the critical zero line, can never be conclusively proven.
Sieve Interpretation
Overlaying all waves acts as an extended Sieve of Eratosthenes:
Sieve grid — waves canceling composites, primes emerging as gaps (frequencies before primes are revealed)
3D Tubular Model
The flat waves twist into a three-dimensional tube spiraling from the central zero-point, resembling gun-
barrel rifling or expanding pond ripples. Precise least common multiple (LCM) calculations give measurable
lengths for repeating segments.
Example: frequencies section 7 - 17 overlapping waves originate and aligned at the zero point.
3D tubular model — waves spiraling outward from the origin. (blue lines represent the #3 frequency)
Prime numbers, orange (3-d) and black (2-d), may represent a melodic binary language.
Perhaps one day prime numbers will be used in cutting-edge communication methods.
Riemann Connection
The waves oscillate around a common zero line.
This visual harmony echoes the behaviour of Riemann zeta function zeros on the critical line Re(s) = 1/2.
The explicit formula links primes to zeta zeros:
ψ(x) = x - ∑_{ρ} x^ρ/ρ - log(2π) - (1/2)log(1 - x^{-2})
(where ρ are non-trivial zeros).
Open Possibilities and Speculation
These unchanging, predictable frequency patterns invite further exploration:
- Longer alignments and their ties to pi digit biases.
- Potential bridges to harmonic models in physics or consciousness.
All ideas remain speculative — awaiting deeper computation and verification.
The primes surprise with order hidden in the spaces between.