Page 96 - Mathematics Class - IX

PROJECT 5

AIM

Generation of Pythagorean triplets.

PYTHAGOREAN TRIPLET
A Pythagorean triplet consists of three positive integers a, b and c such that a + b = c . 3, 4 and 5 is a
2
2
2
Pythagorean triplet.
TYPES OF PYTHAGOREAN TRIPLET
Primitive

Non-primitive
Primitive Pythagorean Triplet: A Pythagorean triplet (a, b, c) is said to be primitive, when a, b, c are coprimes
to each other i.e. HCF (a, b, c) = 1.

Non-primitive Pythagorean Triplet: The Pythagorean triplet in which a, b, and c are not coprimes.

GENERATING PRIMITIVE PYTHAGOREAN TRIPLET
We know that, c is always odd and only one of a and b is odd, so if we choose a as even, ∆ will be even and if we
set a as odd, ∆ will be odd. We illustrate this method by the following example.

Let us consider a = 2 × 3 × 11 = 792. We show the different cases in the following table.
2
3
Table 1: Primitive triplets for a = 792

∆ b (a, b, c)
2 3 11 = 2 156815 (792, 156815, 156817)
0
1 0
2 3 11 = 162 1855 (792, 1855, 2017)
1 4
0
2 3 11 = 242 1175 (792, 1175, 1417)
1 0
2
2 3 11 = 32 9785 (792, 9785, 9817)
5 0
0
2 3 11 = 2592 ---- ----
5 4
0
2 3 11 = 3872 ---- ----
2
5 0
2 3 11 = 19602 ---- ----
1 4
2
2 3 11 = 313632 ---- ----
2
5 4
Here, the last four values of ∆ are not possible because of a > ∆. So, only four primitive triplets can be generated
for a = 792.
Thus, all possible primitive triplets for a given number can be generated.

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