Fungrim home page

Fungrim entry: e922c4

gcd ⁣(a,b)=min{ax+by:xZ  and  yZ  and  ax+by1}\gcd\!\left(a, b\right) = \min \left\{ a x + b y : x \in \mathbb{Z} \;\mathbin{\operatorname{and}}\; y \in \mathbb{Z} \;\mathbin{\operatorname{and}}\; a x + b y \ge 1 \right\}
Assumptions:aZ  and  bZ  and  (a0  or  b0)a \in \mathbb{Z} \;\mathbin{\operatorname{and}}\; b \in \mathbb{Z} \;\mathbin{\operatorname{and}}\; \left(a \ne 0 \;\mathbin{\operatorname{or}}\; b \ne 0\right)
TeX:
\gcd\!\left(a, b\right) = \min \left\{ a x + b y : x \in \mathbb{Z} \;\mathbin{\operatorname{and}}\; y \in \mathbb{Z} \;\mathbin{\operatorname{and}}\; a x + b y \ge 1 \right\}

a \in \mathbb{Z} \;\mathbin{\operatorname{and}}\; b \in \mathbb{Z} \;\mathbin{\operatorname{and}}\; \left(a \ne 0 \;\mathbin{\operatorname{or}}\; b \ne 0\right)
Definitions:
Fungrim symbol Notation Short description
GCDgcd ⁣(a,b)\gcd\!\left(a, b\right) Greatest common divisor
MinimumminxSf(x)\mathop{\min}\limits_{x \in S} f(x) Minimum value of a set or function
ZZZ\mathbb{Z} Integers
Source code for this entry:
Entry(ID("e922c4"),
    Formula(Equal(GCD(a, b), Minimum(Set(Add(Mul(a, x), Mul(b, y)), For(Tuple(x, y)), And(Element(x, ZZ), Element(y, ZZ), GreaterEqual(Add(Mul(a, x), Mul(b, y)), 1)))))),
    Variables(a, b),
    Assumptions(And(Element(a, ZZ), Element(b, ZZ), Or(NotEqual(a, 0), NotEqual(b, 0)))))

Topics using this entry

Copyright (C) Fredrik Johansson and contributors. Fungrim is provided under the MIT license. The source code is on GitHub.

2020-04-08 16:14:44.404316 UTC