A pair of Bézout coefficients ( in fact the ones that are minimal in absolute value ) can be computed by the extended Euclidean algorithm.

A fast way to determine whether two numbers are coprime is given by the Euclidean algorithm.

As a generalization of this, following easily from Euclidean algorithm in base n > 1:

These can be found by applying the extended Euclidean algorithm.

It considers the connection between perfect numbers and Mersenne primes, the infinitude of prime numbers, Euclid's lemma on factorization ( which leads to the fundamental theorem of arithmetic on uniqueness of prime factorizations ), and the Euclidean algorithm for finding the greatest common divisor of two numbers.

* Euclidean algorithm

* Extended Euclidean algorithm

Tarski proved that his axiomatic formulation of elementary Euclidean geometry is consistent and complete in a certain sense: there is an algorithm that, for every proposition, can be shown either true or false.

This generalized Euclidean algorithm can be put to many of the same uses as Euclid's original algorithm in the ring of integers: in any Euclidean domain, one can apply the Euclidean algorithm to compute the greatest common divisor of any two elements.

An arbitrary PID has much the same " structural properties " of a Euclidean domain ( or, indeed, even of the ring of integers ), but knowing an explicit algorithm for Euclidean division, and thus also for greatest common divisor computation, gives a concreteness which is useful for algorithmic applications.

In mathematics, the Euclidean algorithm, or Euclid's algorithm, is an efficient method for computing the greatest common divisor ( GCD ) of two integers, also known as the greatest common factor ( GCF ) or highest common factor ( HCF ).

The earliest surviving description of the Euclidean algorithm is in Euclid's Elements ( c. 300 BC ), making it one of the oldest numerical algorithms still in common use.

If implemented using remainders of Euclidean division rather than subtractions, Euclid's algorithm computes the GCD of large numbers efficiently: it never requires more division steps than five times the number of digits ( base 10 ) of the smaller integer.

The Euclidean algorithm is based on the principle that the greatest common divisor of two numbers does not change if the smaller number is subtracted from the larger number.

Bézout's lemma is a consequence of the Euclidean division defining property, namely that the division by a nonzero integer b has a remainder strictly less than | b |.

For example a Euclidean straight line has no width, but any real drawn line will.

Euclidean geometry has two fundamental types of measurements: angle and distance.

The Hutchinsonian niche is defined more technically as a " Euclidean hyperspace whose dimensions are defined as environmental variables and whose size is a function of the number of values that the environmental values may assume for which an organism has positive fitness.

This approach brings the tools of algebra and calculus to bear on questions of geometry, and has the advantage that it generalizes easily to Euclidean spaces of more than three dimensions.

Once the Euclidean plane has been described in this language, it is actually a simple matter to extend its concept to arbitrary dimensions.

So, given an integral domain R, it is often very useful to know that R has a Euclidean function: in particular, this implies that R is a PID.

A straight line, for instance, is self-similar but not fractal because it lacks detail, is easily described in Euclidean language, has the same Hausdorff dimension as topological dimension, and is fully defined without a need for recursion.

Space, in this construction, still has the ordinary Euclidean geometry.

In integration theory, specifying a measure allows one to define integrals on spaces more general than subsets of Euclidean space ; moreover, the integral with respect to the Lebesgue measure on Euclidean spaces is more general and has a richer theory than its predecessor, the Riemann integral.

Principal ideal domains are thus mathematical objects which behave somewhat like the integers, with respect to divisibility: any element of a PID has a unique decomposition into prime elements ( so an analogue of the fundamental theorem of arithmetic holds ); any two elements of a PID have a greatest common divisor ( although it may not be possible to find it using the Euclidean algorithm ).

A ( topological ) surface is a nonempty second countable Hausdorff topological space in which every point has an open neighbourhood homeomorphic to some open subset of the Euclidean plane E < sup > 2 </ sup >.

* The space is a cube with Euclidean metric ; the figures include cubes of the same size as the space, with colors or patterns on the faces ; the automorphisms of the space are the 48 isometries ; the figure is a cube of which one face has a different color ; the figure has a symmetry group of 8 isometries, there are 6 equivalence classes of 8 isometries, for 6 isomorphic versions of the figure.

From a Euclidean space perspective, the universe has three dimensions of space and one of time.

This metric has only two undetermined parameters: an overall length scale R that can vary with time, and a curvature index k that can be only 0, 1 or − 1, corresponding to flat Euclidean geometry, or spaces of positive or negative curvature.

In mathematics, physics, and engineering, a Euclidean vector ( sometimes called a geometric or spatial vector, or — as here — simply a vector ) is a geometric object that has a magnitude ( or length ) and direction and can be added to other vectors according to vector algebra.

Every rational number / has two closely related expressions as a finite continued fraction, whose coefficients can be determined by applying the Euclidean algorithm to.

Moreover, it has the same number of points as a square of area 1, as a cube of volume 1, and even as an unbounded n-dimensional Euclidean space ( see Space filling curve ).

It has a standard inner product, making it a Euclidean space.

In the Euclidean case, equality occurs only if the triangle has a 180 ° angle and two 0 ° angles, making the three vertices collinear, as shown in the bottom example.

For over two thousand years, the adjective " Euclidean " was unnecessary because no other sort of geometry had been conceived.

His claim seems to have been based on Euclidean presuppositions, because no logical contradiction was present.

The concept has been generalized to differential manifolds of arbitrary dimension embedded in a Euclidean space.

Mathematical applications required geometry of four or more dimensions ; the close scrutiny of the foundations of the traditional Euclidean geometry had revealed the independence of the parallel postulate from the others, and non-Euclidean geometry had been born.

The austere beauty of Euclidean geometry has been seen by many in western culture as a glimpse of an otherworldly system of perfection and certainty.

Then, by placing an arbitrary metric g on a given smooth manifold M and evolving the metric by the Ricci flow, the metric should approach a particularly nice metric, which might constitute a canonical form for M. Suitable canonical forms had already been identified by Thurston ; the possibilities, called Thurston model geometries, include the three-sphere S < sup > 3 </ sup >, three-dimensional Euclidean space E < sup > 3 </ sup >, three-dimensional hyperbolic space H < sup > 3 </ sup >, which are homogeneous and isotropic, and five slightly more exotic Riemannian manifolds, which are homogeneous but not isotropic.

Under certain technical assumptions, it has been shown that a Euclidean QFT can be Wick-rotated into a Wightman QFT.

Most of the work on isoperimetric problem has been done in the context of smooth regions in Euclidean spaces, or more generally, in Riemannian manifolds.

The occasion of the proof by Hassler Whitney of the embedding theorem for smooth manifolds is said ( rather surprisingly ) to have been the first complete exposition of the manifold concept precisely because it brought together and unified the differing concepts of manifolds at the time: no longer was there any confusion as to whether abstract manifolds, intrinsically defined via charts, were any more or less general than manifold extrinsically defined as submanifolds of Euclidean space.

The Euclidean division of polynomials has been the object of specific developments.

Euclid described a line as " breadthless length ", and introduced several postulates as basic unprovable properties from which he constructed the geometry, which is now called Euclidean geometry to avoid confusion with other geometries which have been introduced since the end of nineteenth century ( such as non-Euclidean geometry, projective geometry, and affine geometry ).

In an axiomatic formulation of Euclidean geometry, such as that of Hilbert ( Euclid's original axioms contained various flaws which have been corrected by modern mathematicians ), a line is stated to have certain properties which relate it to other lines and points.

A particular case that has been studied is that in which the linear operator is an isometry M of the hypersphere ( written S < sup > 3 </ sup >) represented within four-dimensional Euclidean space:

Around 1850 the general concept of Euclidean space hadn't been developed — but linear equations in variables were well-understood.

" Hence this axiom could have been named " Congruence is Euclidean.