Pythagorean triple
$$a^2 + b^2 = c^2$$
Euclid's formula:
$$ m,n \in N, m > n $$ $$ a = m^2 - n^2, b = 2mn, c = m^2 + n^2$$ $$ => a^2 + b^2 = c^2$$
Trivial proof.
Primitive pythagorean triples are ones that cannot be reduced (think $ k.a^2 + k.b^2 = k.c^2 $). The triple is primitive if $a$, $b$ and $c$ are coprime (ie they share no common divisors except 1). An example of a primitive pythagorean triple is (3,4,5).
The pythagorean triples generated using Euclid's formula are primitive iff $m$ and $n$ are coprime and $m-n$ is odd.
Clearly this second point is true because if both $m$ and $n$ were even, then all three terms will be even when squared, so $k$ could equal two so the triple would not be primitive.
Similar if they are not coprime, you could obvs divide every term.
$$a^2 + b^2 = c^2$$
Euclid's formula:
$$ m,n \in N, m > n $$ $$ a = m^2 - n^2, b = 2mn, c = m^2 + n^2$$ $$ => a^2 + b^2 = c^2$$
Trivial proof.
Primitive pythagorean triples are ones that cannot be reduced (think $ k.a^2 + k.b^2 = k.c^2 $). The triple is primitive if $a$, $b$ and $c$ are coprime (ie they share no common divisors except 1). An example of a primitive pythagorean triple is (3,4,5).
The pythagorean triples generated using Euclid's formula are primitive iff $m$ and $n$ are coprime and $m-n$ is odd.
Clearly this second point is true because if both $m$ and $n$ were even, then all three terms will be even when squared, so $k$ could equal two so the triple would not be primitive.
Similar if they are not coprime, you could obvs divide every term.
