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Mathematics
Euler's product formula
Prime numbers are key to Euler's product formula, \(\sum_{k=1}^{\infty}\frac{1}{k^s}=\prod_{p \;\; \mathrm{prime}}\frac{1}{1-p^{-s}}\), which establishes equality between an infinite sum and product for fixed \(s>1\).
Mathematics
The closed-form expression for the Fibonacci number
The \(k\)th Fibonacci number can be written in closed form as \(F(k) = \frac{\varphi^k - (-\frac{1}{\varphi})^k}{\sqrt{5}}\), where \(\varphi\) is the celebrated golden ratio.
Physics
The Schrödinger equation
The Schrödinger equation, \(i \hbar \frac{\partial \Psi}{\partial t} = \hat H \Psi\), is a partial differential equation for the wave function \(\Psi\) of quantum mechanics.