Integer triangle

An integer triangle or integral triangle is a triangle all of whose side lengths are integers. A rational triangle is one whose side lengths are rational numbers; any rational triangle can be rescaled by the lowest common denominator of the sides to obtain a similar integer triangle, so there is a close relationship between integer triangles and rational triangles.

Sometimes other definitions of the term rational triangle are used: Carmichael (1914) and Dickson (1920) use the term to mean a Heronian triangle (a triangle with integral or rational side lengths and area); Conway and Guy (1996) define a rational triangle as one with rational sides and rational angles measured in degrees—the only such triangles are rational-sided equilateral triangles.

General properties for an integer triangle

Integer triangles with given perimeter

Any triple of positive integers can serve as the side lengths of an integer triangle as long as it satisfies the triangle inequality: the longest side is shorter than the sum of the other two sides. Each such triple defines an integer triangle that is unique up to congruence. So the number of integer triangles (up to congruence) with perimeter p is the number of partitions of p into three positive parts that satisfy the triangle inequality. This is the integer closest to

p

2

/

48

{\displaystyle p^{2}/48}

when p is even and to

(

p

+

3

)

2

/

48

{\displaystyle (p+3)^{2}/48}

when p is odd. It also means that the number of integer triangles with even numbered perimeters

p

=

2

n

{\displaystyle p=2n}

is the same as the number of integer triangles with odd numbered perimeters

p

=

2

n

−

3.

{\displaystyle p=2n-3.}

Thus there is no integer triangle with perimeter 1, 2 or 4, one with perimeter 3, 5, 6 or 8, and two with perimeter 7 or 10. The sequence of the number of integer triangles with perimeter p, starting at

p

=

1

,

{\displaystyle p=1,}

is:

0, 0, 1, 0, 1, 1, 2, 1, 3, 2, 4, 3, 5, 4, 7, 5, 8, 7, 10, 8 ...