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In mathematics, the floor function is the function that takes as input a real number x, and gives as output the greatest integer less than or equal to x, denoted ⌊x⌋ or floor(x). Similarly, the ceiling function maps x to the least integer greater than or equal to x, denoted ⌈x⌉ or ceil(x). [1]
Many mathematical problems have been stated but not yet solved. These problems come from many areas of mathematics, such as theoretical physics, computer science, algebra, analysis, combinatorics, algebraic, differential, discrete and Euclidean geometries, graph theory, group theory, model theory, number theory, set theory, Ramsey theory, dynamical systems, and partial differential equations.
In mathematics, an integer-valued function is a function whose values are integers.In other words, it is a function that assigns an integer to each member of its domain.. The floor and ceiling functions are examples of integer-valued functions of a real variable, but on real numbers and, generally, on (non-disconnected) topological spaces integer-valued functions are not especially useful.
Goldbach’s Conjecture. One of the greatest unsolved mysteries in math is also very easy to write. Goldbach’s Conjecture is, “Every even number (greater than two) is the sum of two primes ...
Specifically, he considered functions of the form = + (), where a 0, b 0, ..., a P − 1, b P − 1 are rational numbers which are so chosen that g(n) is always an integer. The standard Collatz function is given by P = 2, a 0 = 1 / 2 , b 0 = 0, a 1 = 3, b 1 = 1. Conway proved that the problem
These are the points = + / where can be any nonzero integer; the zeta function can be extended to these values too by taking limits (see [[Dirichlet eta function#Landau's problem with ζ(s) = η(s)/0 and solutions|Dirichlet eta function § Landau's problem with ζ(s) = η(s)/0 and solutions]]), giving a finite value for all values of s with ...
Rectangular function; Floor function: Largest integer less than or equal to a given number. Ceiling function: Smallest integer larger than or equal to a given number. Sign function: Returns only the sign of a number, as +1, −1 or 0. Absolute value: distance to the origin (zero point)
Frobenius coin problem with 2-pence and 5-pence coins visualised as graphs: Sloping lines denote graphs of 2x+5y=n where n is the total in pence, and x and y are the non-negative number of 2p and 5p coins, respectively.