What happens when you keep extending primes into progressively larger bases · LCM(1,…,n) · primorials · the prime number theorem disguised as arithmetic
WP45 established that base 12 = LCM(1,2,3,4) is the minimal base that terminates halves, thirds, and quarters. But minimal is not complete. One third terminates cleanly in base 12; one fifth does not. To capture one fifth, you need base 60 = LCM(1,2,3,4,5). To capture one seventh, you need base 420.
There is a ladder. Each rung is a base. Each rung is reached by adding one new prime. The question is: what do you find as you climb?
The ladder is the sequence of values taken by LCM(1, 2, …, n) as n increases. The LCM jumps only when n is a prime or a prime power — those are the only values that add a new factor not already present.
The pattern is already visible in the first eight rungs: jumps at primes and prime powers; flat stretches in between. Base 12 and base 60 are the two rungs that sit at the bottom of the ladder — sufficient for everyday arithmetic, reachable in human time, physically manageable as a counting system. Above 60, each rung is an abstraction.
For each rung, the table below shows which simple fractions 1/n now terminate — highlighted in teal where a fraction first becomes exact at that rung.
Each new prime added to the base buys coverage of exactly the fractions \(1/p\), \(1/p^2\), \(1/p^3\), … for that prime \(p\). The value of this purchase falls as \(p\) grows, for a simple reason: fractions with large denominators appear less often in practical computation.
Adding 2 to the base gives you one half — the most useful single fraction in arithmetic. Adding 3 gives you one third — the second most useful. Adding 5 gives one fifth — useful for decimal compatibility and for pentagons. Adding 7 gives one seventh — useful for musical temperament, ancient calendars, and little else in everyday measurement. Adding 11 gives one eleventh. Adding 13 gives one thirteenth.
The practical sweet spot — the point where adding the next prime costs more base than it earns in coverage — sits somewhere between base 12 and base 60, depending on what you are measuring. The Babylonians identified base 60 as their sweet spot roughly 4000 years ago and were right.
There is a deeper structure to the ladder. Define \(\psi(n) = \log \text{LCM}(1, 2, \ldots, n)\). This is the Chebyshev function — one of the central objects in analytic number theory. The prime number theorem, in its most elementary form, is the statement:
In plain language: the logarithm of the LCM of the first \(n\) integers grows like \(n\). Equivalently, \(\text{LCM}(1, \ldots, n) \approx e^n\). Each unit step up the integer sequence multiplies the LCM by approximately \(e \approx 2.718\).
The prime ladder is the prime number theorem made visible. Each rung is a data point. The asymptotic growth rate of the rungs is the statement that the primes are, on average, as common as they are. Gauss conjectured this in 1792. Hadamard and de la Vallée Poussin proved it in 1896. Every time you compute LCM(1,…,n), you are computing a value whose growth rate encodes a deep truth about the distribution of primes.
There is no top. The ladder is infinite. For any base \(B\), no matter how many primes it contains, there exists a prime \(p > B\) such that \(1/p\) does not terminate in base \(B\). This follows directly from the infinitude of primes — a proof Euclid gave around 300 BCE, and which has not needed revision since.
More precisely: the density of \(B\)-smooth numbers — integers all of whose prime factors are prime factors of \(B\) — tends to zero as the integers grow. No matter how many primes you include in your base, the fraction of all integers whose reciprocal terminates in that base approaches zero. This is not a failure of the base; it is a theorem about the primes.
What you find at the top of the ladder, then, is not completeness. It is the prime number theorem. The ladder is a staircase encoding the distribution of primes, and climbing it is equivalent to sampling that distribution. The steps come faster at first (2, 3, 5 are close together) and then spread out — but never stop.
The G6 Crystal lives in a rank-6 space. The root lattice is E₆; the Weyl group has order 51840; the minimal vectors number 72. The rank is 6.
LCM(1, 2, 3, 4, 5, 6) = 60. Rank 6 is the rung at which the LCM reaches 60 — the Babylonian base, the base of hours and degrees. No coincidence is claimed; the correspondence is noted.
What is structurally true: the six-fold divisibility of 60 — its factorisation 2²·3·5 — matches the three fundamental symmetries of the G6 Crystal exactly:
The sixth rung of the prime ladder is where the crystal lives. This is not a derivation — it is an observation. But it is the kind of observation that points toward a derivation.
The LCM sequence climbs by prime powers as well as primes. There is a faster version — the primorial sequence — that takes only the primes themselves, one at a time:
Primorials are the simplest bases that cover exactly the first \(k\) primes. They grow faster than the LCM sequence (which pauses at prime powers). Their growth rate is exactly \(e^{p_k}\), where \(p_k\) is the \(k\)-th prime — again encoding the PNT.
The primorials have a second life in number theory: consecutive primorials are the denominators that appear in Mertens' theorem and the Euler product for the Riemann zeta function at 1. Every primorial is the denominator of a partial product over primes. The ladder, climbed via primorials, is the zeta function made concrete.
Five open problems in number theory — among the oldest and most resistant — each sit at a specific position on the prime ladder. None has been solved by changing base. But each becomes more legible when you know which rung it inhabits.
Take any positive integer. If even, halve it. If odd, multiply by three and add one. The conjecture: you always reach 1. Simple to state; open since 1937. Erdős said mathematics was not yet ready for it.
The operations split on base-2 parity — even and odd is a binary question. The ×3 is a ternary step. These two primes together put the conjecture's natural home at rung 3 of the ladder: base 6 = LCM(1,2,3) = 2×3, where both operations are simultaneously legible in a single numeral. Researchers have studied Collatz in base 6 and in base 3. No proof has emerged. What the ladder reveals is the structure of the question, not its answer.
Every even integer greater than 2 is the sum of two primes. Stated in a letter to Euler in 1742. Euler replied that he was certain it was true and could not prove it. Two hundred and eighty years have passed without a general proof.
Goldbach lives at every rung of the ladder simultaneously — it is a statement about all primes, not about any particular prime. Changing base does not illuminate it because its structure is analytic, not arithmetic. The Hardy-Littlewood circle method is the most powerful tool available; it works over the complex numbers, not over a counting base. This is the conjecture most immune to the ladder.
There are infinitely many pairs of primes that differ by 2: (3,5), (5,7), (11,13), (17,19), (29,31)… The pairs thin out as numbers grow larger but, the conjecture says, never stop.
In base 12, all primes greater than 3 end in 1, 5, 7, or B — the four residue classes coprime to 12. Every twin prime pair (p, p+2) with p > 3 has p ending in 5 or B in dozenal. The structure is visible at rung 4. Zhang's 2013 proof that prime gaps are bounded — that there exist infinitely many primes differing by at most 70 million, later tightened — is the closest approach. It does not use a base-change argument. The twin prime conjecture remains open.
For every integer n ≥ 2, the fraction 4/n can be written as the sum of three unit fractions: 4/n = 1/x + 1/y + 1/z. This is Egyptian fraction arithmetic — the question of which denominators allow clean decomposition.
In base 12, 4/n terminates in one digit whenever n divides 12 — the six divisors of 12 give immediate solutions. This is the same structure WP45 built for terminating fractions generally. The Erdős-Straus conjecture has been verified computationally for very large n. No general proof exists. It is the conjecture most directly connected to the dozenal work of WP45, and the one whose partial cases become most transparent at rung 4.
For coprime integers a + b = c, the conjecture bounds how large c can be relative to rad(abc) — the product of the distinct prime factors of a, b, and c. The radical is the primorial-adjacent object: strip a number to its distinct primes and multiply them. This is the floor of the ladder — the rung structure itself, read downward.
The ABC conjecture is a statement about the relationship between a number and its prime skeleton. Mochizuki published a claimed proof in 2012 — over five hundred pages of inter-universal Teichmüller theory, restructuring arithmetic at a level that is not base change but something more fundamental. The mathematical community has not reached consensus. The conjecture remains open. What the ladder offers is a way to see what rad(abc) is: the product of distinct primes, the signature of which rungs a number touches.
The question at the start of WP45 was: what do you find when you keep extending primes? The answer, now fully stated: