Fermat numberIn mathematics, a Fermat number, named after Pierre de Fermat, the first known to have studied them, is a positive integer of the form where n is a non-negative integer. The first few Fermat numbers are: 3, 5, 17, 257, 65537, 4294967297, 18446744073709551617, ... . If 2k + 1 is prime and k > 0, then k itself must be a power of 2, so 2k + 1 is a Fermat number; such primes are called Fermat primes. , the only known Fermat primes are F0 = 3, F1 = 5, F2 = 17, F3 = 257, and F4 = 65537 ; heuristics suggest that there are no more.
Prime numberA prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only ways of writing it as a product, 1 × 5 or 5 × 1, involve 5 itself. However, 4 is composite because it is a product (2 × 2) in which both numbers are smaller than 4.
Square rootIn mathematics, a square root of a number x is a number y such that ; in other words, a number y whose square (the result of multiplying the number by itself, or ) is x. For example, 4 and −4 are square roots of 16 because . Every nonnegative real number x has a unique nonnegative square root, called the principal square root, which is denoted by where the symbol "" is called the radical sign or radix. For example, to express the fact that the principal square root of 9 is 3, we write .
Complex numberIn mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted i, called the imaginary unit and satisfying the equation ; every complex number can be expressed in the form , where a and b are real numbers. Because no real number satisfies the above equation, i was called an imaginary number by René Descartes. For the complex number , a is called the , and b is called the . The set of complex numbers is denoted by either of the symbols or C.
Square (algebra)In mathematics, a square is the result of multiplying a number by itself. The verb "to square" is used to denote this operation. Squaring is the same as raising to the power 2, and is denoted by a superscript 2; for instance, the square of 3 may be written as 32, which is the number 9. In some cases when superscripts are not available, as for instance in programming languages or plain text files, the notations x^2 (caret) or x**2 may be used in place of x2. The adjective which corresponds to squaring is quadratic.
Quadratic equationIn algebra, a quadratic equation () is any equation that can be rearranged in standard form as where x represents an unknown value, and a, b, and c represent known numbers, where a ≠ 0. (If a = 0 and b ≠ 0 then the equation is linear, not quadratic.) The numbers a, b, and c are the coefficients of the equation and may be distinguished by respectively calling them, the quadratic coefficient, the linear coefficient and the constant coefficient or free term.
Complex planeIn mathematics, the complex plane is the plane formed by the complex numbers, with a Cartesian coordinate system such that the x-axis, called the real axis, is formed by the real numbers, and the y-axis, called the imaginary axis, is formed by the imaginary numbers. The complex plane allows a geometric interpretation of complex numbers. Under addition, they add like vectors.
Mersenne primeIn mathematics, a Mersenne prime is a prime number that is one less than a power of two. That is, it is a prime number of the form Mn = 2n − 1 for some integer n. They are named after Marin Mersenne, a French Minim friar, who studied them in the early 17th century. If n is a composite number then so is 2n − 1. Therefore, an equivalent definition of the Mersenne primes is that they are the prime numbers of the form Mp = 2p − 1 for some prime p. The exponents n which give Mersenne primes are 2, 3, 5, 7, 13, 17, 19, 31, .
Quadratic functionIn mathematics, a quadratic polynomial is a polynomial of degree two in one or more variables. A quadratic function is the polynomial function defined by a quadratic polynomial. Before the 20th century, the distinction was unclear between a polynomial and its associated polynomial function; so "quadratic polynomial" and "quadratic function" were almost synonymous. This is still the case in many elementary courses, where both terms are often abbreviated as "quadratic".
Fermat's Last TheoremIn number theory, Fermat's Last Theorem (sometimes called Fermat's conjecture, especially in older texts) states that no three positive integers a, b, and c satisfy the equation an + bn = cn for any integer value of n greater than 2. The cases n = 1 and n = 2 have been known since antiquity to have infinitely many solutions. The proposition was first stated as a theorem by Pierre de Fermat around 1637 in the margin of a copy of Arithmetica. Fermat added that he had a proof that was too large to fit in the margin.
Imaginary numberAn imaginary number is a real number multiplied by the imaginary unit i, which is defined by its property i2 = −1. The square of an imaginary number bi is −b2. For example, 5i is an imaginary number, and its square is −25. By definition, zero is considered to be both real and imaginary. Originally coined in the 17th century by René Descartes as a derogatory term and regarded as fictitious or useless, the concept gained wide acceptance following the work of Leonhard Euler (in the 18th century) and Augustin-Louis Cauchy and Carl Friedrich Gauss (in the early 19th century).
Pierpont primeIn number theory, a Pierpont prime is a prime number of the form for some nonnegative integers u and v. That is, they are the prime numbers p for which p − 1 is 3-smooth. They are named after the mathematician James Pierpont, who used them to characterize the regular polygons that can be constructed using conic sections. The same characterization applies to polygons that can be constructed using ruler, compass, and angle trisector, or using paper folding. Except for 2 and the Fermat primes, every Pierpont prime must be 1 modulo 6.
Constructible polygonIn mathematics, a constructible polygon is a regular polygon that can be constructed with compass and straightedge. For example, a regular pentagon is constructible with compass and straightedge while a regular heptagon is not. There are infinitely many constructible polygons, but only 31 with an odd number of sides are known. Some regular polygons are easy to construct with compass and straightedge; others are not.
Quadratic formulaIn elementary algebra, the quadratic formula is a formula that provides the solution(s) to a quadratic equation. There are other ways of solving a quadratic equation instead of using the quadratic formula, such as factoring (direct factoring, grouping, AC method), completing the square, graphing and others. Given a general quadratic equation of the form whose discriminant is positive, with x representing an unknown, with a, b and c representing constants, and with a ≠ 0, the quadratic formula is: where the plus–minus symbol "±" indicates that the quadratic equation has two solutions.
Methods of computing square rootsMethods of computing square roots are numerical analysis algorithms for approximating the principal, or non-negative, square root (usually denoted , , or ) of a real number. Arithmetically, it means given , a procedure for finding a number which when multiplied by itself, yields ; algebraically, it means a procedure for finding the non-negative root of the equation ; geometrically, it means given two line segments, a procedure for constructing their geometric mean. Every real number except zero has two square roots.
Quadratic integerIn number theory, quadratic integers are a generalization of the usual integers to quadratic fields. Quadratic integers are algebraic integers of degree two, that is, solutions of equations of the form x2 + bx + c = 0 with b and c (usual) integers. When algebraic integers are considered, the usual integers are often called rational integers. Common examples of quadratic integers are the square roots of rational integers, such as , and the complex number i = , which generates the Gaussian integers.
Complex logarithmIn mathematics, a complex logarithm is a generalization of the natural logarithm to nonzero complex numbers. The term refers to one of the following, which are strongly related: A complex logarithm of a nonzero complex number , defined to be any complex number for which . Such a number is denoted by . If is given in polar form as , where and are real numbers with , then is one logarithm of , and all the complex logarithms of are exactly the numbers of the form for integers .
Doubling the cubeDoubling the cube, also known as the Delian problem, is an ancient geometric problem. Given the edge of a cube, the problem requires the construction of the edge of a second cube whose volume is double that of the first. As with the related problems of squaring the circle and trisecting the angle, doubling the cube is now known to be impossible to construct by using only a compass and straightedge, but even in ancient times solutions were known that employed other tools.
Root of unityIn mathematics, a root of unity, occasionally called a de Moivre number, is any complex number that yields 1 when raised to some positive integer power n. Roots of unity are used in many branches of mathematics, and are especially important in number theory, the theory of group characters, and the discrete Fourier transform. Roots of unity can be defined in any field. If the characteristic of the field is zero, the roots are complex numbers that are also algebraic integers.
Straightedge and compass constructionIn geometry, straightedge-and-compass construction – also known as ruler-and-compass construction, Euclidean construction, or classical construction – is the construction of lengths, angles, and other geometric figures using only an idealized ruler and a pair of compasses. The idealized ruler, known as a straightedge, is assumed to be infinite in length, have only one edge, and no markings on it. The compass is assumed to have no maximum or minimum radius, and is assumed to "collapse" when lifted from the page, so may not be directly used to transfer distances.
