Euclid's Formula and Properties of Pythagorean Triples

generators

m > n

m

3

m and n coprime

n

2

one even, one odd

m	n	2 m - 2 n	2mn	2 m + 2 n
3	2	5	12	13

{a, b, c}:	2 5	+	2 12		2 13
{5,12,13}	25	+	144		169

primitive?

True

common factor:

1 

2

1

(c - a)(c - b)/2 is perfect square	True
(c - even leg) and (c - odd leg)/2 are both squares	True
at most one of a, b, c is square	True
area of triangle cannot be a square or twice a square	True
exactly one of a, b is odd	True
exactly one of a, b is divisible by 3	True
exactly one of a, b is divisible by 4	True
exactly one of a, b, c is divisible by 5	True
all prime factors of c have the form 4n + 1	True
a, b, c are relatively prime	True

A Pythagorean triple consists of three integers

a

,

b

and

c

, such that

2

a

+

2

b

=

2

c

. If

a

,

b

and

c

are relatively prime, they form a primitive Pythagorean triple. Euclid’s formula generates a Pythagorean triple for every choice of positive integers

m

and

n

. Adjust the sliders to change the generating integers and see which of the tests are satisfied by the triple generated. Checkboxes apply additional restrictions.