Miscellaneous

Miscellaneous Calculation Solver

Golden Ratio: $ϕ = \frac{1 + \sqrt{5}}{2}$
Fibonacci Sequence: $F_{n} = F_{n - 1} + F_{n - 2}$
Lucas Sequence: $L_{n} = L_{n - 1} + L_{n - 2}$
Pascal's Triangle: $(\binom{n}{k}) = (\binom{n - 1}{k - 1}) + (\binom{n - 1}{k})$
Catalan Numbers: $C_{n} = \frac{1}{n + 1} (\binom{2 n}{n})$
Stirling Numbers: $S (n, k)$
Bell Numbers: $B_{n} = \sum_{k = 0}^{n} S (n, k)$
Partition Function: $p (n)$
Riemann Hypothesis: $ζ (s) = 0$ for $Re (s) = \frac{1}{2}$
Poincaré Conjecture: Every simply connected, closed 3-manifold is homeomorphic to the 3-sphere.
P vs NP Problem: Is P equal to NP?
Twin Prime Conjecture: Infinitely many primes $p$ such that $p + 2$ is also prime.
Goldbach's Conjecture: Every even integer > 2 is the sum of two primes.
Collatz Conjecture: For any positive integer $n$ , the sequence $n, f (n), f (f (n)), \dots$ eventually reaches 1.
Four Color Theorem: Any map can be colored with four colors.
Banach-Tarski Paradox: A solid ball can be decomposed into a finite number of pieces and reassembled into two identical copies.
Cantor's Diagonal Argument: The real numbers are uncountable.
Zorn's Lemma: Every partially ordered set has a maximal element.
Axiom of Choice: Every set of non-empty sets has a choice function.
Continuum Hypothesis: There is no set whose cardinality is strictly between that of the integers and the real numbers.