Solutions to famous mathematical questions with equations
Executive summary
A brief tour of celebrated equations shows how some mathematical questions are neatly solved by compact formulas — for example, the Pythagorean relation a^2 + b^2 = c^2 and Einstein’s E = mc^2 — while other “famous” problems remain open or only partially tamed by equations, as with Navier–Stokes or P vs NP (the latter still debated in popular sources) [1] [2] [3] [4].
1. Classical geometry made concrete: the Pythagorean equation
The most elementary solution-bearing equation is the Pythagorean theorem, which states that for any right triangle the square of the hypotenuse equals the sum of the squares of the legs, a^2 + b^2 = c^2, and this relation has been taught since antiquity and continues to serve as a cornerstone of geometry [1] [5].
2. Unity of constants: Euler’s identity as a ‘solution’ to aesthetic questions
Euler’s identity e^{iπ} + 1 = 0 is celebrated for tying together five fundamental constants and for resolving an aesthetic question about how disparate mathematical objects — e, i, π, 1, 0 — coherently interact in a single equation, a fact often invoked in profiles of the most beautiful equations [2].
3. Equations that solved scientific puzzles: E = mc^2 and Shannon’s capacity
Einstein’s E = mc^2 distilled a physical insight — mass–energy equivalence — into a simple equation that answered long-standing questions about the relationship between mass and energy [1], while Claude Shannon’s channel-capacity formula C = B · log2(1 + S/N) (often written in variants) provided a concrete bound on how much information can be transmitted over a noisy channel and so solved a central problem in communications theory [6].
4. Partial solutions and grand open problems: Navier–Stokes and P vs NP
Not all famous mathematical questions admit neat closed-form solutions; the Navier–Stokes equations give an extraordinarily useful system for fluid flow whose practical — often numerical — solutions power engineering simulations, yet the existence of smooth, global solutions in three dimensions remains an open million-dollar Clay Prize problem [3] [7], and computational complexity’s P vs NP question is discussed as the kind of statement that, if resolved (for example by a proof of P = NP), would rewrite theory and practice but remains unresolved and controversial in public-facing sources [4].
5. Ancient diophantine puzzles and algebraic strategies: Pell equations and the rise of calculus
Equations that ask for integer solutions — exemplified by forms now called Pell equations — have occupied thinkers from Brahmagupta and Bhāskara II through later European mathematicians and admit algorithmic methods for producing integer solutions rather than single closed-form identities [8]; similarly, the advent of calculus — credited independently to Newton and Leibniz — transformed problems about slopes, areas and limits into equations and methods that “solved” whole classes of previously intractable problems by providing differential and integral formalisms used across science [9].
6. What counts as a ‘solution’ and the hidden agendas of lists of equations
Popular lists and magazine pieces — such as Ian Stewart’s accounts or “equations that changed the world” compilations — frame certain formulas as decisive solutions because they tie disparate ideas together or deliver practical engineering power [3] [10] [7], but those lists reflect choices and narratives: some highlight elegance (Scientific American’s picks), others emphasize historical impact (Business Insider, Live Science), and none replace the technical literature that distinguishes closed-form proofs, numerical approximations, algorithmic procedures, and open conjectures [11] [1] [10].