PrimeMind · Structural atlas · two units

Two-Unit Lab

What can be built from 2 and 3 alone? Take every power of each — whole or fractional steps — and combine them: multiply (2^a·3^b), add (2^a + 3^b), subtract (2^a − 3^b, with absolute values if you want the negatives folded up). The set is drawn four ways: the exponent lattice, a grid of (a, b) where each cell is a number and lights up if it is prime; the number line, the whole set placed out of any order; the waveform of the set in log-time with its spectrum; and the two-prime zeta — what a world with only the primes 2 and 3 rings at, against the real zeros.

Build the set

rule

a runs from 0 to

b runs from 0 to

exponent step (1 = whole powers; 0.5 = half powers; 0.25 …)

keep values up to (up to 1e12; primality is tested below 1e10)

first unit p

second unit q

absolute values (fold negatives up) keep only whole numbers (with fractional exponents most values are not whole)

The exponent lattice — across: a (power of the first unit) · up: b (power of the second) · each cell is one number · lit: prime · amber: a prime that is one of a twin pair · dim: composite · dark: not a whole number

With the multiply rule nothing lights except the two axes (2^a and 3^b are prime only when the exponent is 1): a product of 2s and 3s is never any other prime, which is the exact statement that primes cannot be built from smaller primes by multiplication. With the add and subtract rules the lattice lights in scattered cells — some primes can be reached, most cannot: below 2,000, the rule 2^a ± 3^b reaches 60 of the 303 primes and misses 243, starting with 53, 71, 103, 107, 109. That scattering is a real and unsolved corner of number theory (Pillai's problem asks how often 2^a − 3^b can take a given value).

The number line — the whole set placed out of any order (log scale across; height = which rule made it)

On a log scale the products 2^a·3^b form a perfect two-dimensional lattice pressed onto a line: the point sits at a·log 2 + b·log 3. Because log 2 and log 3 are incommensurable, the pattern never repeats but never clusters randomly either — it is quasi-periodic, the one-dimensional shadow of a grid. Sums and differences break the lattice.

The waveform — the set as waves in log-time: Σ cos(t·log n)/√n over the set, with its spectrum (one line per member) and the real zeros marked

window: from t =

to t =

samples across the window

overlay the primes-only part (white) overlay the composites-only part (amber)

scroll on the wave to zoom at the cursor · drag to pan · the window recomputes at full resolution

Structure finder — the spikes, named: the lattice's resonances (Bragg peaks) for the current step and units

draw the predicted resonance lines on the wave (cyan dashed: multiples of 2π ÷ (step·log p); green dashed: multiples of 2π ÷ (step·log q))

What a two-prime world rings at — the zeros of 1/ζ restricted to the units: (1 − p^−s)(1 − q^−s), against the real zeros

With only two primes, zeta's Euler product has two factors, and 1/ζ is (1 − 2^−s)(1 − 3^−s). Its zeros are two evenly spaced combs: t = 2πk/log 2 (every 9.06) and t = 2πm/log 3 (every 5.72), and they sit on the line Re s = 0, not ½. Add a third prime and you get a third comb; the true zeros — irregular, repelling, on the line ½ — appear only when every prime is present, because it is the interference of infinitely many incommensurable combs that produces them. That is the precise sense in which the zeros are a property of the whole set of primes and of no finite part of it.