Pythagorean Triplets
Pythagorean triplets are sets of three positive integers, a, b, and c, that satisfy the Pythagorean theorem: a^2 + b^2 = c^2. In other words, these triplets represent the sides of a right-angled triangle.

Characteristics of Pythagorean Triplets
- The most well-known Pythagorean triplet is (3, 4, 5) where 3^2 + 4^2 = 5^2.
- Pythagorean triplets can be scaled up by multiplying all three numbers by a common factor.
- There are infinitely many Pythagorean triplets, and they can be generated using various formulas and techniques.

Primitive Pythagorean Triplets
Primitive Pythagorean triplets are those in which the greatest common divisor of a, b, and c is 1. These triplets cannot be scaled up by a common factor.

Generating Pythagorean Triplets
- One way to generate Pythagorean triplets is using the Euclid's formula: a = m^2 - n^2, b = 2mn, c = m^2 + n^2, where m and n are positive integers and m > n.
- Another method is to use the Pythagorean theorem itself to check for triplets by iterating through possible values of a, b, and c.

Applications of Pythagorean Triplets
- Pythagorean triplets have applications in geometry, physics, and engineering for calculating distances, angles, and solving various problems involving right-angled triangles.
- They are also used in cryptography and number theory for generating secure keys and solving mathematical puzzles.
In conclusion, Pythagorean triplets are a fascinating set of integers that have rich mathematical properties and practical applications across various fields.