  Fundamental mathematical constants like $e$ and $\pi$ are ubiquitous in
diverse fields of science, from abstract mathematics to physics, biology and
chemistry. For centuries, new formulas relating fundamental constants have been
scarce and usually discovered sporadically. Here we propose a novel and
systematic approach that leverages algorithms for deriving mathematical
formulas for fundamental constants and help reveal their underlying structure.
Our algorithms find dozens of well-known as well as previously unknown
continued fraction representations of $\pi$, $e$, Catalan's constant, and
values of the Riemann zeta function. Two example conjectures found by our
algorithm and so far unproven are: \begin{equation*} \frac{24}{\pi^2} = 2 +
7\cdot 0\cdot 1+ \frac{8\cdot1^4}{2 + 7\cdot 1\cdot 2 + \frac{8\cdot2^4}{2 +
7\cdot 2\cdot 3 + \frac{8\cdot3^4}{2 + 7\cdot 3\cdot 4 +
\frac{8\cdot4^4}{..}}}} \quad\quad,\quad\quad \frac{8}{7 \zeta(3)} = 1\cdot 1 -
\frac{1^6}{3\cdot 7 - \frac{2^6}{5\cdot 19 - \frac{3^6}{7\cdot 37 -
\frac{4^6}{..}}}} \end{equation*} We present two algorithms that proved useful
in finding conjectures: a Meet-In-The-Middle (MITM) algorithm and a Gradient
Descent (GD) tailored to the recurrent structure of continued fractions. Both
algorithms are based on matching numerical values and thus they conjecture
formulas without providing proofs and without requiring prior knowledge on any
underlying mathematical structure. This approach is especially attractive for
constants for which no mathematical structure is known, as it reverses the
conventional approach of sequential logic in formal proofs. Instead, our work
supports a different approach for research: algorithms utilizing numerical data
to unveil mathematical structures, thus trying to play the role of intuition of
great mathematicians of the past, providing leads to new mathematical research.
