the sides of such a triangle form a Heronian triple for . Every Pythagorean triple is a Heronian triple, because at least one of the legs , must be even in a Pythagorean triple, so the area ab/2 is an integer. Not every Heronian triple is a Pythagorean triple, however, as the example with area 24 shows. If is a Heronian triple, so is where is any positive integer; its area will be the integer that is times the integer area of the triangle. The Heronian triple is primitive provided a, b, c are setwise coprime. (With primitive Pythagorean triples the stronger statement that they are pairwise coprime also applies, but with primitive Heronian triangles the stronger statement does not always hold true, such as with .) Here are a few of the simplest primitive Heronian triples that are not Pythagorean triples: (4, 13, 15) with area 24 (3, 25, 26) with area 36 (7, 15, 20) with area 42 (6, 25, 29) with area 60 (11, 13, 20) with area 66 (13, 14, 15) with area 84 (13, 20, 21) with area 126 By Heron's formula, the extra condition for a triple of positive integers with to be Heronian is that or equivalently be a nonzero perfect square divisible by 16. Application to cryptography Primitive Pythagorean triples have been used in cryptography as random sequences and for the generation of keys. See also Boolean Pythagorean triples problem Congruum Diophantus II.VIII Eisenstein triple Euler brick Heronian triangle Hilbert's theorem 90 Integer triangle Modular arithmetic Nonhypotenuse number Plimpton 322 Pythagorean prime Pythagorean quadruple Quadric Tangent half-angle formula Trigonometric identity Notes References External links Clifford Algebras and Euclid's Parameterization of Pythagorean triples Curious Consequences of a Miscopied Quadratic Discussion of Properties of Pythagorean triples, Interactive Calculators, Puzzles and Problems Generating Pythagorean Triples Using Arithmetic Progressions Interactive Calculator for Pythagorean Triples The negative Pell equation and Pythagorean triples Parameterization of Pythagorean Triples by a single triple of polynomials Pythagorean Triples and the Unit Circle, chap. 2–3, in "A Friendly Introduction to Number Theory" by Joseph H. Silverman, 3rd ed., 2006, Pearson Prentice Hall, Upper Saddle River, NJ, Pythagorean Triples at cut-the-knot Interactive Applet showing unit circle relationships to Pythagorean Triples Pythagorean Triplets The Remarkable Incircle of a Triangle Solutions to Quadratic Compatible Pairs in relation to Pythagorean Triples Theoretical properties of the Pythagorean Triples and connections to geometry The Trinary Tree(s) underlying Primitive