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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 \(s>1\).

The binomial theorem, making use of the well-known binomial coefficients, allows expressing \((x+y)^n\) as a sum: \((x+y)^n=\sum_{k=0}^{n} \binom{n}{k} x^k y^{n-k}\).

Factorials apply only to nonnegative integers but can be extended to complex numbers with positive real parts using the Gamma function, \(\Gamma(n) = \int_0^\infty e^{-x}x^{n-1}dx\).