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Set-Theoretic Concepts and Number Theory
Published in Nirdosh Bhatnagar, Introduction to Wavelet Transforms, 2020
As mentioned earlier, the extended Euclidean algorithm implicitly uses the Euclidean algorithm. If two positive integers a and b are given such that b ≤ a, and gcd(a,b)=d, the extended Euclidean algorithm expresses the greatest common divisor as d=(αa+βb), where α, β ∈ ℤ. This result is called Bezout’s theorem for integers. It is named after Étíenne Bézout (1730–1753). The extended Euclidean algorithm is not described in this chapter. These concepts are best illustrated via an example.
Number Theory
Published in Paul L. Goethals, Natalie M. Scala, Daniel T. Bennett, Mathematics in Cyber Research, 2022
The most common use of the extended Euclidean algorithm is the computation of inverses modulo n. Note, however, that only one of the Bézout coefficients is necessary for this purpose, as demonstrated by the proof of the lemma.
The uniqueness for inverse discrete exterior transmission eigenvalue problems in spherically symmetric media
Published in Applicable Analysis, 2021
In this section, we illustrate Theorems 3.3 and 3.4 in the above sections with some explicit examples. Let From (14), we have and Since and are coprime, we have by using the extended Euclidean algorithm for polynomial, where and
The power of the snake: number theory with Python
Published in International Journal of Mathematical Education in Science and Technology, 2022
One of the first substantial results in number theory is that the greatest common divisor (gcd) is always a linear combination of its inputs. Students should learn how to find the coefficients of such a linear combination with the Euclidean Algorithm (Burton, 2010, p. 26). But besides doing this somewhat tedious computation by hand, there is also the power of code. After my students have mastered the Euclidean Algorithm with small numbers, I give them the following ‘extended gcd’ function, adapted from Extended Euclidean Algorithm (2021):
The inverse discrete transmission eigenvalue problem for absorbing media
Published in Inverse Problems in Science and Engineering, 2018
and can be computed by the extended Euclidean algorithm for polynomial. Then doing the Euclidean division of polynomial by we have