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Euler number – g.c.d.
Euler number
The transcendental number e, approximately 2.71828.... It may be defined as the limit
or as the infinite sum
The function ex is called the exponential function, and has the property that it is its own derivative.
Euler phi function
Given a natural number n, this function returns the number of integers between 1 and n which are relatively prime to n.
even function
A real-valued function y = f(x) is even if f(x) = f(–x) for all x in the domain of f. The graph of an even function in the Cartesian plane is symmetric with respect to the y-axis.
Cf. odd function.
exp
See exponential function.
exponent
Also called index, a number or expression applying to another number or expression, called the base, and indicating that the base is to be “raised to the power of” the exponent. For integers, this corresponds to the operation of repeated self-multiplication the number of times specified by the exponent.
The exponent is written to the right of, and superscripted to, the base.
Cf. rational exponent, laws of exponents.
exponential function
The function ex. We say that an expression is expontiated when it is used as an exponent on e.
More generally, any function which operates on its argument by placing it as an exponent may be called an exponential function. Example: f(x) = 2x.
Cf. Euler number.
factor
The factors of a natural number n are those whole numbers which divide it evenly. Since every number is divisible by itself and 1, neither 1 nor n is considered a proper factor of n.
More generally, if an expression can be written as a product of other expressions, then those other expressions are its factors. For instance, a polynomial with rational coefficients may always be factored into a product of first and/or second degree polynomial factors.
A number (or other expression) with no proper factors is called prime.
Graph Theory: A spanning subgraph of a given graph having at least one edge. In many contexts, it is interesting to determine whether some graph G can be decomposed into the edge-disjoint union of factors with some prescribed property. Such a decomposition is called a factorization of G. Often, the property in question is regularity of degree k. In this case, the factors are called k-factors, and the factorization a k-factorization. If G has a k-factorization, it is called k-factorable.
factorial
An operation on natural numbers, denoted n! and given by n! = 1 × 2 × 3 × ... × n. Thus 3! = 1 × 2 × 3 = 6, and 4! = 24. By convention, 0! = 1 and 1! = 1.
factorization
The process of factoring, that is, of finding proper factors.
fallacy
ARTICLE
An unjustified step in logic, or incorrect form of reasoning, leading to an invalid conclusion. See the article for a complete description.
Fermat’s Last Theorem
The Diophantine equation x n + y n = z n has no non-trivial solutions in integers x, y, and z for any n greater than 2. (There are infinitely many solutions in integers for n = 2, and these are called the Pythagorean triples.) Fermat penned this theorem in the margin of his copy of the Arithmetica of Diophantus of Alexandria, and added, “I have discovered a remarkable proof of this theorem, which unfortunately this margin is too small to contain.” He died without ever writing down the proof, and the theorem remained unproved for 300 years, until Andrew Wiles presented a proof at a Cambridge lecture in 1992.
Fermat’s Little Theorem
For any integer n and prime number p that does not divide n, n p - 1 is congruent to 1 modulo p.
Cf. Euler phi function.
Fermat test
A test for primality based on the converse of Fermat's little theorem:
the Fermat test for n base b is that the congruence bp-1  1 modulo n should hold. For prime n, this is always provided that
b and n are coprime; however there
are also some composite numbers which will pass the test. For example, 341 = 11 × 31 is composite, yet 2340 1 modulo 341, so we call 2 a Fermat pseudoprime to the base 2.
Even worse, Carmichael numbers are pseudoprimes for the Fermat test and any base.
Fortunately, pseudoprimes and Carmichael numbers are rare compared to primes.
Such tests are thus probabilistic, in that numbers passing the test are likely but not certain to be prime.
Fibonacci sequence
ARTICLE
The sequence discovered by the medieval mathematician Fibonacci, and described in his book Liber Abaci. See the article for a complete description.
figure
General: A drawing, picture, or illustration, usually accompanying a text description. Also synonymous with digit or numeral.
Geometry: A graphical (visual) representation of points, lines, curves, surfaces, solids, or regions. The word “figure” may also be used to refer to the abstract object or set of points thus represented.
Cf. plane figure.
finite set
A set X is finite if there is a natural number n (possibly 0) such that X can be said to have exactly n elements. More formally, a set is finite if it is not bijective with any proper subset of itself.
function
Given a binary relation R on sets A and B, we say that the R is a function if each element of A is paired with at most one element of B. In this case we call A the domain set, and we call B the range set. If f is such a function and the ordered pair (x,y) is an element of the function, we typically write f(x) = y.
More generally, if R is an n-place relation, then R is a function of n-1 variables if each (n-1)-tuple is matched with at most one element of the range set (i.e., the nth set of the Cartesian product on which the relation is defined). If in addition there is at most one element of the domain set matched with any given element of the range set, then the function is called “one-to-one” or injective. If the function maps at least one element of the domain set to every element of its range set, then it is called “onto” or surjective. A function which is both injective and surjective is called bijective. Functions with range in the real numbers or complex numbers are called real-valued or complex-valued respectively.
Fundamental Theorem of Algebra
The field of complex numbers is
algebraically closed, that is, any polynomial with complex coefficients has a complex root.
Fundamental Theorem of Arithmetic
ARTICLE
Every natural number is either prime or may be decomposed uniquely into prime factors.
Fundamental Theorem of Calculus
Intitively, that the integral and derivative are inverse operatorations on functions. The fundamental theorem is commonly expressed in either one of two formal guises:
Cf. Riemann integral.
g.c.d.
Abbreviation for greatest common divisor.
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