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Graphs of functions commonly used in the analysis of algorithms, showing the number of operations versus input size for each function. The following tables list the computational complexity of various algorithms for common mathematical operations.
The number of connected simple cubic graphs on 4, 6, 8, 10, ... vertices is 1, 2, 5, 19, ... (sequence A002851 in the OEIS).A classification according to edge connectivity is made as follows: the 1-connected and 2-connected graphs are defined as usual.
*: The narrow class number is also 1 (see related sequence A003655 in OEIS). Despite what would appear to be the case for these small values, not all prime numbers that are congruent to 1 modulo 4 appear on this list, notably the fields Q(√ d) for d = 229 and d = 257 both have class number greater than 1 (in fact equal to 3 in both cases). [3]
The Catalogue of Triangle Cubics is an online resource containing detailed information about more than 1200 cubic curves in the plane of a reference triangle. [1] The resource is maintained by Bernard Gibert. Each cubic in the resource is assigned a unique identification number of the form "Knnn" where "nnn" denotes three digits.
Because of the factorization (2n + 1)(n 2 + n + 1), it is impossible for a centered cube number to be a prime number. [3] The only centered cube numbers which are also the square numbers are 1 and 9, [4] [5] which can be shown by solving x 2 = y 3 + 3y, the only integer solutions being (x,y) from {(0,0), (1,2), (3,6), (12,42)}, By substituting a=(x-1)/2 and b=y/2, we obtain x^2=2y^3+3y^2+3y+1.
The number of elements of a finite field is called its order or, sometimes, its size. A finite field of order q {\displaystyle q} exists if and only if q {\displaystyle q} is a prime power p k {\displaystyle p^{k}} (where p {\displaystyle p} is a prime number and k {\displaystyle k} is a positive integer).
The Fibonacci cube of order n is the simplex graph of the complement graph of an n-vertex path graph. [2] That is, each vertex in the Fibonacci cube represents a clique in the path complement graph, or equivalently an independent set in the path itself; two Fibonacci cube vertices are adjacent if the cliques or independent sets that they represent differ by the addition or removal of a single ...
Cubic reciprocity is a collection of theorems in elementary and algebraic number theory that state conditions under which the congruence x 3 ≡ p (mod q) is solvable; the word "reciprocity" comes from the form of the main theorem, which states that if p and q are primary numbers in the ring of Eisenstein integers, both coprime to 3, the congruence x 3 ≡ p (mod q) is solvable if and only if ...