The dataset viewer is not available for this split.
Error code: StreamingRowsError
Exception: CastError
Message: Couldn't cast
total_cfs: int64
converged: int64
matches: int64
match_details: list<item: struct<type: string, label: string, P: list<item: int64>, Q: list<item: int64>, constant: (... 60 chars omitted)
child 0, item: struct<type: string, label: string, P: list<item: int64>, Q: list<item: int64>, constant: string, re (... 48 chars omitted)
child 0, type: string
child 1, label: string
child 2, P: list<item: int64>
child 0, item: int64
child 3, Q: list<item: int64>
child 0, item: int64
child 4, constant: string
child 5, relation: string
child 6, digits: int64
child 7, cf_value: string
elapsed_seconds: double
slow_converger_details: list<item: null>
child 0, item: null
new_matches: int64
total_hits: int64
total_checked: int64
slow_convergers: int64
to
{'total_hits': Value('int64'), 'total_checked': Value('int64'), 'slow_convergers': Value('int64'), 'new_matches': Value('int64'), 'matches': List(Value('null')), 'slow_converger_details': List(Value('null')), 'elapsed_seconds': Value('float64')}
because column names don't match
Traceback: Traceback (most recent call last):
File "/src/services/worker/src/worker/utils.py", line 99, in get_rows_or_raise
return get_rows(
^^^^^^^^^
File "/src/libs/libcommon/src/libcommon/utils.py", line 272, in decorator
return func(*args, **kwargs)
^^^^^^^^^^^^^^^^^^^^^
File "/src/services/worker/src/worker/utils.py", line 77, in get_rows
rows_plus_one = list(itertools.islice(ds, rows_max_number + 1))
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
File "/usr/local/lib/python3.12/site-packages/datasets/iterable_dataset.py", line 2690, in __iter__
for key, example in ex_iterable:
^^^^^^^^^^^
File "/usr/local/lib/python3.12/site-packages/datasets/iterable_dataset.py", line 2227, in __iter__
for key, pa_table in self._iter_arrow():
^^^^^^^^^^^^^^^^^^
File "/usr/local/lib/python3.12/site-packages/datasets/iterable_dataset.py", line 2251, in _iter_arrow
for key, pa_table in self.ex_iterable._iter_arrow():
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
File "/usr/local/lib/python3.12/site-packages/datasets/iterable_dataset.py", line 494, in _iter_arrow
for key, pa_table in iterator:
^^^^^^^^
File "/usr/local/lib/python3.12/site-packages/datasets/iterable_dataset.py", line 384, in _iter_arrow
for key, pa_table in self.generate_tables_fn(**gen_kwags):
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
File "/usr/local/lib/python3.12/site-packages/datasets/packaged_modules/json/json.py", line 289, in _generate_tables
self._cast_table(pa_table, json_field_paths=json_field_paths),
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
File "/usr/local/lib/python3.12/site-packages/datasets/packaged_modules/json/json.py", line 124, in _cast_table
pa_table = table_cast(pa_table, self.info.features.arrow_schema)
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
File "/usr/local/lib/python3.12/site-packages/datasets/table.py", line 2272, in table_cast
return cast_table_to_schema(table, schema)
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
File "/usr/local/lib/python3.12/site-packages/datasets/table.py", line 2218, in cast_table_to_schema
raise CastError(
datasets.table.CastError: Couldn't cast
total_cfs: int64
converged: int64
matches: int64
match_details: list<item: struct<type: string, label: string, P: list<item: int64>, Q: list<item: int64>, constant: (... 60 chars omitted)
child 0, item: struct<type: string, label: string, P: list<item: int64>, Q: list<item: int64>, constant: string, re (... 48 chars omitted)
child 0, type: string
child 1, label: string
child 2, P: list<item: int64>
child 0, item: int64
child 3, Q: list<item: int64>
child 0, item: int64
child 4, constant: string
child 5, relation: string
child 6, digits: int64
child 7, cf_value: string
elapsed_seconds: double
slow_converger_details: list<item: null>
child 0, item: null
new_matches: int64
total_hits: int64
total_checked: int64
slow_convergers: int64
to
{'total_hits': Value('int64'), 'total_checked': Value('int64'), 'slow_convergers': Value('int64'), 'new_matches': Value('int64'), 'matches': List(Value('null')), 'slow_converger_details': List(Value('null')), 'elapsed_seconds': Value('float64')}
because column names don't matchNeed help to make the dataset viewer work? Make sure to review how to configure the dataset viewer, and open a discussion for direct support.
Ramanujan Machine — GPU Formula Discovery Results
GPU-accelerated search for new continued fraction formulas for mathematical constants, inspired by Raayoni et al. (2024).
Part of the bigcompute.science project. AI-audited, not peer-reviewed.
Key Findings (Updated 2026-04-07)
- 586 billion equal-degree polynomial CFs exhausted (v1 kernel, degrees 1-8) — zero new transcendental formulas discovered.
- 7,030 "transcendental hits" were double-precision false positives — all disproven via 100-digit PSLQ verification (verify_hits.py).
- Only 20 confirmed formulas — all classical: Euler's e, Brouncker's 4/pi, Leibniz pi/4, 1/ln(2).
- Root cause identified: the v1 kernel forced deg(a_n) = deg(b_n), but every known CF formula for transcendental constants has deg(b_n) ≈ 2 × deg(a_n). Equal-degree CFs converge super-exponentially to algebraic numbers and cannot produce new transcendental formulas.
- v2 kernel built (ramanujan_v2.cu) with independent degrees for numerator and denominator polynomials. Validated on (1,2) regime — 48 confirmed transcendental formulas at 120-200 digit precision.
v1 Results (Equal-Degree Search)
| Degree | Range | Candidates | Constants Found |
|---|---|---|---|
| 1-3 | up to [-40,40] | ~282B | sqrt(2), sqrt(5), phi only |
| 4 | [-7,7] | 577B | sqrt(2) only |
| 5 | [-5,5] | 3.1T | sqrt(2) only |
| 6-8 | [-2,2] to [-4,4] | ~60T | sqrt(2) only |
| Total | 586B+ | Zero new transcendental |
PSLQ Verification (2026-04-07)
- 7,030 claimed transcendental matches → all false positives at 100-digit precision
- 20 confirmed formulas → all classical, previously known
- Additional tests: deeper CF evaluation (depth 5000), expanded constant library (30 constants incl. MZVs, Glaisher, Khinchin), rational coefficients — all negative
v2 Results (Asymmetric-Degree Search, In Progress)
| Config (deg_a, deg_b) | Range | Candidates | Converged | Confirmed |
|---|---|---|---|---|
| (1, 2) | [-10,10] | 4.1M | 3M (73%) | 48 transcendental (classical) |
| (2, 4) | [-6,6] | 816M | 521M (64%) | In progress |
The v2 kernel also saves all converged-but-unmatched CFs to enable offline multi-constant PSLQ scanning (pslq_scan.py).
Method
v1 (deprecated)
For polynomial pairs (P, Q) of the same degree with bounded integer coefficients, evaluate the generalized CF to double precision (500 terms), then match against 10 base constants + 29 compound expressions.
v2 (current)
CF = a(0) + b(1) / (a(1) + b(2) / (a(2) + ...)) where a(n) has degree d_a and b(n) has degree d_b independently. Target: d_b ≈ 2 × d_a (the "productive zone" for transcendental constants). GPU evaluates at double precision; survivors verified via CPU PSLQ at 100+ digits.
Understanding This Data
The Ramanujan Machine project tries to discover new mathematical formulas by brute force: generate billions of continued fraction expressions and check whether any of them equal known constants like pi, e, or zeta(3).
The key lesson from this dataset: the polynomial degree structure matters more than the search volume. 586 billion equal-degree candidates produced nothing, while 4 million asymmetric-degree candidates immediately re-derived classical formulas. The productive zone is deg(numerator) ≈ 2 × deg(denominator), matching the theoretical insight from Raayoni et al.'s Conservative Matrix Field framework.
Source
- Code: ramanujan_v2.cu (v2), ramanujan_gpu.cu (v1)
- Verification: verify_hits.py, pslq_scan.py
- Experiment: bigcompute.science
Citation
@misc{humphreys2026ramanujan,
author = {Humphreys, Cahlen and Claude (Anthropic)},
title = {Ramanujan Machine: GPU-Accelerated CF Formula Discovery},
year = {2026},
publisher = {Hugging Face},
url = {https://huggingface.co/datasets/cahlen/ramanujan-machine-results}
}
Human-AI collaborative work. AI-audited against published literature. Not independently peer-reviewed. CC BY 4.0.
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