Shoulders of Giants

Swiss mathematician who essentially founded the theory of continued fractions, discovered the number e, and contributed to every branch of mathematics known in his time. His work on CF convergents is the algorithmic backbone of our Zaremba verification and Ramanujan Machine experiments.

The "Prince of Mathematics" who introduced class numbers of quadratic forms, the Gauss map for continued fractions, and foundational work on quadratic fields. Our class number computation extends tables he began in 1801, now reaching 30 billion discriminants on GPU hardware he could not have imagined.

German mathematician whose question-mark function ?(x) provides a remarkable bijection between rationals ordered by CF structure and rationals ordered by size. We compute the full multifractal singularity spectrum of this function — to our knowledge, the first numerical computation of f(α) for the Minkowski measure.

German mathematician who introduced the concept of fractional dimension that bears his name. We compute dim_H(E_A) for over a million continued fraction Cantor sets. Hausdorff was persecuted under the Nazi regime; in 1942, rather than be sent to a concentration camp, he and his wife took their own lives.

American mathematician whose 1901 classification of subgroups of SL₂(𝔽ₚ) we use to prove that the Zaremba semigroup acts transitively on (ℤ/pℤ)² for every prime p. This eliminates local obstructions to the conjecture — a result possible only because Dickson's century-old classification is complete.

Indian mathematician who, largely self-taught, discovered thousands of formulas for mathematical constants — many involving continued fractions of extraordinary beauty. The Ramanujan Machine experiment is named for his approach: systematic discovery of formulas through pattern recognition, now accelerated by GPU computation.

Irish-American mathematician who co-developed the Murnaghan-Nakayama rule for computing characters of the symmetric group. This recursive formula — walking border strips on Young diagrams — powers our GPU computation of 26.4 billion Kronecker coefficients for S₃₀.

Polish mathematician who conjectured that for every integer d, there exists a coprime fraction a/d whose continued fraction partial quotients are all bounded by 5. This conjecture, posed in 1972 in the context of quasi-Monte Carlo integration, has driven decades of work in analytic number theory and is the central focus of our Zaremba experiments.

British mathematician and philosopher who founded Ramsey theory at age 25 with a single paper, then died at 26 of hepatitis. His 1930 theorem — that sufficiently large structures must contain ordered substructures — launched an entire field. The question R(5,5)=? remains open nearly a century later.

French mathematician who, with Lenstra, formulated the Cohen-Lenstra heuristics predicting the distribution of class groups of number fields. Our computation of 30 billion class numbers tests these predictions at a scale never before possible, revealing that convergence to the predicted 75% is far slower than expected.

Dutch mathematician who co-formulated the Cohen-Lenstra heuristics and co-invented the LLL lattice reduction algorithm. His probabilistic model for class groups — that groups appear with frequency proportional to 1/|Aut(G)| — is one of the most elegant conjectures in algebraic number theory.

Belgian mathematician and Fields Medalist (1994). With Kontorovich, proved that Zaremba's conjecture holds for a density-1 set of integers using the circle method and property (τ) for thin groups. His spectral gap techniques underpin our entire approach to making the conjecture effective.

American mathematician at Rutgers. With Bourgain, proved the density-1 result for Zaremba's conjecture. His work on thin groups, expanders, and the spectral theory of automorphic forms provides the theoretical framework that our GPU computations aim to make effective.