Zaremba Exception Hierarchy: 27 → 2 → 0
The Finding
The 27 exceptions to full Zaremba density with A={1,2,3} (verified to 10^{10}) have a precise hierarchical structure:
| Digit set | Exceptions | Which ones |
|---|---|---|
| A={1,2,3} | 27 | all <= 6234 |
| A={1,2,3,4} | 2 | d=54, d=150 only |
| A={1,2,3,4,5} | 0 | Zaremba’s conjecture |
Adding digit 4 resolves 25 of the 27 exceptions. The remaining 2 (d=54, d=150) require digit 5.
The CF Splitting Identity
An important subtlety: d=6 appears in the 27 exceptions for A={1,2,3} because its canonical CF representation 5/6 = [0; 1, 5] uses digit 5. However, the non-canonical form [0; 1, 4, 1] = 5/6 uses only digits {1, 4}. The continued fraction identity
[0; a_1, …, a_k] = [0; a_1, …, a_k - 1, 1]
allows the last quotient to be split, potentially reducing the maximum digit by 1 at the cost of one extra term. This is why d=6 is covered by A={1,2,3,4} even though the standard CF of 5/6 needs digit 5.
The Two Stubborn Exceptions
d=54: every coprime fraction a/54 has a partial quotient of at least 5 in its continued fraction expansion. No representation — canonical or non-canonical — avoids digit 5.
d=150: best CF is 29/150 = [0; 5, 5, 1, 4]. No splitting resolves the double-5 structure.
References
- Zaremba, S.K. (1972). “La methode des bons treillis.”
- Bourgain, J. and Kontorovich, A. (2014). “On Zaremba’s conjecture.” Annals of Mathematics.
This work was produced through human-AI collaboration (Cahlen Humphreys + Claude). Not independently peer-reviewed. All code and data open for verification at github.com/cahlen/idontknow.
Why d=54 and d=150 Are Special
Both stubborn exceptions share structural properties:
| d=54 | d=150 | |
|---|---|---|
| Factorization | 2 x 3^3 | 2 x 3 x 5^2 |
| Divisible by 6 | yes | yes |
| Prime power factor | 3^3 | 5^2 |
| GCD(54, 150) | 6 | 6 |
| Best max partial quotient | 5 | 5 |
For d=54, EVERY coprime fraction a/54 has a partial quotient of at least 5. There are 18 coprime residues mod 54, and none of their CFs avoid digit 5. Similarly for d=150 (40 coprime residues, all CFs require digit 5).
These are the only 2 integers in [1, 10^6] where digit 5 is truly unavoidable — making them the “hardest” denominators for Zaremba’s conjecture.