Fungrim entry: e922c4

\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:

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
`GCD`	$\gcd\!\left(a, b\right)$	Greatest common divisor
`Minimum`	$\mathop{\min}\limits_{x \in S} f(x)$	Minimum value of a set or function
`ZZ`	$\mathbb{Z}$	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

Greatest common divisor