Digit-Partitioning Primes and the Alignment Formula
The previous result
In the companion paper, we proved that the repetend alignment of n = 3m equals (2m − 1)/(3m − 1), and that the golden ratio selects p = 3 uniquely through a self-referential algebraic condition.
A natural question follows: does the same formula hold for primes other than 3?
The answer
Yes. The formula
$$\alpha(pm) = \frac{2m - 1}{pm - 1}$$
holds for every prime p satisfying a single condition: p ≤ b + 1, where b is the positional base. We call such primes digit-partitioning.
In base 10, the digit-partitioning primes are exactly 3, 7, and 11.
Why it works
The proof rests on one object: the digit function
$$\delta(r) = \left\lfloor \frac{br}{p} \right\rfloor$$
This is the machine inside long division. Given a remainder r, it produces the next digit. When p ≤ b + 1, the digit function is injective: different remainders always produce different digits.
Injectivity means that at every position of the repetend, no two fractions k/p and 1/p can share the same digit (unless k = 1). Every non-matching fraction contributes zero alignment. The counting argument from the first paper carries through unchanged, with 3 replaced by any digit-partitioning prime p.
The boundary case
When p = b + 1, the digit function simplifies to
$$\delta(r) = r - 1$$
A bijection from {1, ..., b} to {0, ..., b − 1}. Every remainder maps to a unique digit. Every digit is used exactly once. No room to spare.
In base 10, this gives p = 11. The five cosets of ⟨10⟩ in (ℤ/11ℤ)* produce the repetend pairs:
- {1, 10} → 09
- {2, 9} → 18
- {3, 8} → 27
- {4, 7} → 36
- {5, 6} → 45
These are the nines-complement pairs. They partition all ten digits with nothing left over.
Three mechanisms, one condition
In base 10, the three digit-partitioning primes arise through different structural mechanisms:
- p = 3: single-digit repetends (ord₃(10) = 1)
- p = 7: cyclic number 142857, all digits distinct (ord₇(10) = 6 = p − 1)
- p = 11: complement-pair partition (10 ≡ −1 mod 11)
These initially appeared to require separate treatment. The characterization p ≤ b + 1 unifies them. All three are consequences of the injectivity of the digit function.
The golden ratio, universally
Since the formula (2m − 1)/(pm − 1) holds for all digit-partitioning primes, the golden threshold analysis applies uniformly. The alignment limit as m → ∞ is 2/p, which exceeds 1/φ only when
$$p < 2\varphi \approx 3.236$$
Among primes, only p = 3 satisfies this. Across all bases and all digit-partitioning primes, the golden ratio selects p = 3 uniquely. The self-referential characterization, the fourth-power identity τ⁴ = 2 − 3τ, and the factorization (τ² + τ − 1)(τ − 2) = 0 extend without modification.
The digit-partitioning framework reveals that this selection is not an artifact of base 10. It is a universal feature of the alignment formula.
The paper
The full paper, with proofs of the characterization theorem, the universal alignment formula, and the enumeration across bases:
Digit-Partitioning Primes and the Alignment Formula (PDF)
Both papers can be verified computationally using nfield:
./nfield verify # paper 1
./nfield verify2 # paper 2
.:.