Fungrim entry: e922c4

\gcd\!\left(a, b\right) = \min\left(\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\}\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(\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\}\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`	$\gcd\!\left(n, k\right)$	Greatest common divisor
`Minimum`	$\mathop{\min}\limits_{P\left(x\right)} f\!\left(x\right)$	Minimum value of a set or function
`SetBuilder`	$\left\{ f\!\left(x\right) : P\!\left(x\right) \right\}$	Set comprehension
`ZZ`	$\mathbb{Z}$	Integers

Source code for this entry:

Entry(ID("e922c4"),
    Formula(Equal(GCD(a, b), Minimum(SetBuilder(Add(Mul(a, x), Mul(b, y)), 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(Unequal(a, 0), Unequal(b, 0)))))

Topics using this entry

Greatest common divisor