Zaremba Exception Hierarchy: 27 → 2 → 0

The Finding

For the digit set $A=\{1,2,3\}$, the completed CPU/GPU logs in this repository show exactly 27 uncovered denominators through $10^{10}$, all at most $6{,}234$:

$6, 20, 28, 38, 42, 54, 96, 150, 156, 164, 216, 228, 318, 350, 384, 558, 770, 876, 1014, 1155, 1170, 1410, 1870, 2052, 2370, 5052, 6234.$

Adding digit 4 resolves 25 of the 27. The only two remaining uncovered denominators for $A=\{1,2,3,4\}$ are:

$54,\quad 150.$

Adding digit 5 covers both, so the finite-range hierarchy is:

Digit set	Uncovered among the 27	Interpretation
$\{1,2,3\}$	27	Not covered through the completed $10^{10}$ run
$\{1,2,3,4\}$	2	Only $54$ and $150$ remain
$\{1,2,3,4,5\}$	0	All 27 are covered

This is a computational decomposition of a finite list. It does not prove that $A=\{1,2,3\}$ has no further exceptions beyond the completed search range.

Verification

The local CPU reference implementation reproduces the 27 exceptions at $10^6$:

gcc -O3 -o zaremba_density scripts/experiments/zaremba-density/zaremba_density.c -lm
./zaremba_density 1000000 1,2,3

The committed GPU log scripts/experiments/zaremba-density/results/gpu_A123_1e9.log records the same 27 exceptions at $10^9`, and the density phase-transition finding records the completed$10^{10}$ run.

Witnesses After Adding Digit 4

One witness numerator for each resolved denominator is listed below. Each row means $a/d = [0; a_1,\ldots,a_k]$ with every partial quotient in $\{1,2,3,4\}$.

$d$	$a$	Continued fraction
6	5	$[0;1,4,1]$
20	11	$[0;1,1,4,1,1]$
28	17	$[0;1,1,1,1,4,1]$
38	21	$[0;1,1,4,3,1]$
42	23	$[0;1,1,4,1,2,1]$
96	53	$[0;1,1,4,3,2,1]$
156	97	$[0;1,1,1,1,1,4,3,1]$
164	103	$[0;1,1,1,2,4,1,2,1]$
216	137	$[0;1,1,1,2,1,3,4,1]$
228	139	$[0;1,1,1,1,3,1,1,4,1]$
318	197	$[0;1,1,1,1,1,2,4,1,1,1]$
350	207	$[0;1,1,2,4,3,1,2,1]$
384	235	$[0;1,1,1,1,2,1,2,1,4,1]$
558	347	$[0;1,1,1,1,1,4,2,1,3,1]$
770	479	$[0;1,1,1,1,1,4,1,2,1,1,1,1]$
876	559	$[0;1,1,1,3,4,2,2,2,1]$
1014	629	$[0;1,1,1,1,1,2,1,2,2,4,1]$
1155	713	$[0;1,1,1,1,1,1,2,2,4,2,1]$
1170	751	$[0;1,1,1,3,1,4,2,3,1,1]$
1410	877	$[0;1,1,1,1,1,4,1,1,3,1,2,1]$
1870	1159	$[0;1,1,1,1,1,2,2,1,2,4,1,1]$
2052	1265	$[0;1,1,1,1,1,1,4,1,4,1,3,1]$
2370	1441	$[0;1,1,1,1,4,2,1,1,3,4,1]$
5052	3115	$[0;1,1,1,1,1,1,4,3,3,2,2,1]$
6234	3845	$[0;1,1,1,1,1,1,3,1,1,1,2,4,2,1]$

The Two Stubborn Exceptions

For $d=54$ and $d=150$, exhaustive search over coprime numerators shows no representation with all partial quotients in $\{1,2,3,4\}$. Allowing digit 5 gives representations, for example:

$d$	$a$	Continued fraction
54	35	$[0;1,1,1,5,2,1]$
150	91	$[0;1,1,1,1,5,2,1,1]$

These two denominators are therefore the hardest elements of the observed 27-exception list.

References

1. Zaremba, S.K. (1972). “La méthode des bons treillis.” Applications of Number Theory to Numerical Analysis, pp. 39–119.
2. Bourgain, J. and Kontorovich, A. (2014). “On Zaremba’s conjecture.” Annals of Mathematics, 180(1), pp. 137–196.
3. Hensley, D. (1996). “A polynomial time algorithm for the Hausdorff dimension of continued fraction Cantor sets.” Journal of Number Theory, 58(1), pp. 9–45.

---

This work was produced through human-AI collaboration. Not independently peer-reviewed. All code and data are open for verification at github.com/cahlen/idontknow.