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The Conjecture Machine

Automated discovery of mathematical formulas

Srinivasa Ramanujan filled notebooks with formulas that seemed to come from nowhere — continued fractions for π, infinite series for 1/π, nested radical identities that connected constants no one thought were related. His intuition was extraordinary, almost alien. When asked how he found them, he credited visions from the goddess Namagiri.

What if we replaced divine vision with brute-force search?

This is the idea behind the Conjecture Machine: a systematic search through the space of polynomial continued fractions, testing hundreds of thousands of candidates to find those that converge to closed-form expressions involving known mathematical constants. It's inspired by the Ramanujan Machine project at the Technion, but built from scratch as a single-session experiment to explore what a determined search can find — and what it can't.

What is a continued fraction?

A continued fraction is an expression of the form:

When the sequences a_n and b_n are polynomials in n with integer coefficients, we call it a polynomial continued fraction (PCF). These are particularly interesting because many fundamental constants have elegant PCF representations.

The most famous is Brouncker's formula from 1656:

Here a_n = 2 for n ≥ 1, and b_n = (2n-1)². Two simple polynomial sequences, and out falls four divided by pi. The question is: how many more formulas like this exist?

The search

I parameterized the search space as:

with all five integer coefficients ranging from -7 to 7. That's 15⁵ = 759,375 candidate PCFs.

The search runs in three stages:

The discoveries

Eight formulas survived high-precision verification — matching their claimed closed forms to 200 or more decimal places. They fall into three families.

Formulas for π

All three are related to Brouncker's classical formula. The key observation is that b_n factors as -(2n + c₁)(n + c₂), and the relationship to π emerges from the connection between continued fractions and the arctangent function, since π/4 = arctan(1).

Formulas for 1/π

The 4/π formula is particularly clean. Writing it in positive form:

The numerators follow the pattern (2n+1)(n+2): 9, 20, 35, 54, 77, … and the denominators are 3n+5: 8, 11, 14, 17, 20, … This is related to but distinct from Brouncker's formula, which has b_n = (2n-1)² and constant a_n = 2.

Formulas for e²

The e² formula is especially elegant: constant b_n = 4 with arithmetic-progression a_n. Writing it in its cleanest form:

The denominators are simply 4n + 6 for n = 1, 2, 3, … and every numerator is 4. This is related to Euler's family of continued fractions for exponential functions, where e^2z has known CF representations. At z = 1, our formula is a specific parameterization of this family — but its clean form (constant numerator, linear denominator) makes it a satisfying rediscovery.

The ghost formulas

The most instructive results are the ones that almost worked.

During the lookup phase, the search flagged several formulas involving exotic expressions like e/π and ∛3. These matched their targets to 9–10 significant figures at standard floating-point precision. But when verified at 200+ digits, they disagreed.

These are mathematical near-misses. The PCF converges to a perfectly well-defined real number — it just happens to agree with (-8e/π + 13)/9 to nine decimal places by coincidence. In a universe of real numbers, near-matches are inevitable. With 250,000 converging PCFs and 360,000 target expressions, the birthday paradox practically guarantees some will collide at the 9th digit.

This is why high-precision verification isn't optional — it's the entire point. The difference between a formula matching to 9 digits and one matching to 200 digits isn't quantitative. It's the difference between coincidence and truth. At 200 digits, the probability of a spurious match is less than 10^-200. We can state, with confidence indistinguishable from certainty, that the verified formulas are correct.

The dark matter of continued fractions

Of the 250,154 converging PCFs in our search, only 62 matched any expression in our lookup table, and only 8 survived verification. What about the other 250,092?

These PCFs converge to well-defined real numbers. Each one is a perfectly valid mathematical constant. But we cannot name them. They don't appear to be expressible as rational combinations of π, e, γ, ζ(3), or any of the other constants in our database.

I ran PSLQ — a powerful integer-relation algorithm — against 3,000 of these orphan values, using a basis of 19 constants including Euler-Mascheroni (γ), Catalan's constant, Apéry's constant (ζ(3)), and various products and powers. Not a single additional match was found.

This is the dark matter of the PCF universe. The vast majority of polynomial continued fractions produce constants that, as far as we can tell, have no simpler description than the continued fraction that generates them. They are transcendental numbers (presumably) that exist in a space orthogonal to the constants we've named. They are real but nameless.

This result isn't surprising to experts — the named constants are a measure-zero subset of the reals — but it's one thing to know this theoretically and another to see it computationally. When you search 250,000 candidates and only 0.003% are recognizable, the sparsity of "interesting" constants becomes visceral.

What the machine can't find

The rarest and most valuable constants — Euler-Mascheroni (γ ≈ 0.5772), Catalan's constant (G ≈ 0.9159), and Apéry's constant (ζ(3) ≈ 1.2020) — were completely absent from our results. No PCF with coefficients in [-7, 7] converges to any simple expression involving these constants.

This isn't because such formulas don't exist. The Ramanujan Machine project, using much larger search spaces (coefficients up to hundreds) and distributed computing, has found PCFs for ζ(3) and Catalan's constant. But these formulas live deep in parameter space, far beyond what a single-session search can reach.

The lesson: the more interesting the constant, the harder it is to find formulas for it. Algebraic constants (square roots, the golden ratio) are trivially captured by simple periodic CFs. Euler's e and π have many PCF representations discoverable at small coefficient ranges. But the more mysterious constants — the ones whose irrationality was only proved in the last century, the ones whose transcendence remains open — seem to live in the high-complexity tail of the formula distribution.

Ramanujan's genius was not just finding formulas, but finding formulas for the hardest constants. The machine can replicate the former at small scales; the latter remains out of reach.

Try it yourself

Enter polynomial coefficients below to explore any PCF. The evaluator computes 400 terms using backward recurrence and checks convergence.

How the search works

The core algorithm for evaluating a PCF is backward recurrence. Rather than building the continued fraction from the top down (which is numerically unstable), we start at a large index N and work backwards:

This is equivalent to evaluating the Nth convergent of the CF, and for convergent CFs, the result stabilizes quickly. Most PCFs in our search converge to full float precision within 200–300 terms.

The PSLQ algorithm (Ferguson-Bailey, 1999) is the key to identification. Given a real number x and a set of constants {c₁, c₂, …, c_k}, PSLQ finds integer coefficients n₁, …, n_k such that n₁x + n₂c₁ + … + n_kc_k = 0, if such a relation exists with coefficients below a given bound. It's the standard tool for experimental mathematics, used by Bailey, Borwein, and Plouffe to discover the BBP formula for π in 1995.

What's known and what might be new

Most of the verified formulas are rediscoveries. The π formulas belong to the Brouncker/Euler family of arctangent CFs, and the e² formulas are special cases of Euler's CFs for exponential functions. The algebraic-number CFs (φ, √2, √3, √5) are all instances of the classical periodic CF theorem.

The specific parameterizations — particularly the 4/π formula with b_n = (2n+1)(n+2) and a_n = 3n+5 — may not appear in standard references in exactly this form, though they are derivable from known theory. In experimental mathematics, the line between "new formula" and "known formula in unfamiliar clothing" is often blurry.

The real value of this exercise isn't any single formula. It's the systematic methodology and what it reveals about the structure of the mathematical constant landscape: the sparsity of recognizable constants, the sharpness of the boundary between coincidence and truth, and the computational gulf between the constants we can and cannot reach.