Search results
Results from the WOW.Com Content Network
Here the greatest common divisor of 0 and 0 is taken to be 0.The integers x and y are called Bézout coefficients for (a, b); they are not unique.A pair of Bézout coefficients can be computed by the extended Euclidean algorithm, and this pair is, in the case of integers one of the two pairs such that | x | ≤ | b/d | and | y | ≤ | a/d |; equality occurs only if one of a and b is a multiple ...
In arithmetic and computer programming, the extended Euclidean algorithm is an extension to the Euclidean algorithm, and computes, in addition to the greatest common divisor (gcd) of integers a and b, also the coefficients of Bézout's identity, which are integers x and y such that + = (,).
In the case of Bézout's theorem, the general intersection theory can be avoided, as there are proofs (see below) that associate to each input data for the theorem a polynomial in the coefficients of the equations, which factorizes into linear factors, each corresponding to a single intersection point. So, the multiplicity of an intersection ...
By dividing both sides by c/g, the equation can be reduced to Bezout's identity sa + tb = g, where s and t can be found by the extended Euclidean algorithm. [69] This provides one solution to the Diophantine equation, x 1 = s (c/g) and y 1 = t (c/g). In general, a linear Diophantine equation has no solutions, or an infinite number of solutions ...
This is a linear Diophantine equation, related to Bézout's identity. + = + The smallest nontrivial solution in positive integers is 12 3 + 1 3 = 9 3 + 10 3 = 1729.It was famously given as an evident property of 1729, a taxicab number (also named Hardy–Ramanujan number) by Ramanujan to Hardy while meeting in 1917. [1]
Note: "lc" stands for the leading coefficient, the coefficient of the highest degree of the variable. This algorithm computes not only the greatest common divisor (the last non zero r i), but also all the subresultant polynomials: The remainder r i is the (deg(r i−1) − 1)-th subresultant polynomial.
A DEXA scan (dual-energy X-ray absorptiometry) is the “gold standard” for calculating body composition because it’s low cost, low radiation, and very accurate, Dr. Busse says.
Multiplying a and b respectively by the Bézout coefficients for d with respect to a 0 and b 0 gives a polynomial p in aS + bS with constant term d. Then p − d has a zero constant term, and so is a multiple in S of the constant polynomial r, and therefore lies in aS + bS. But then d does as well, which completes the proof.