Pythagorean Triplet:
Pythagorean triplet refers to a set of three positive integers a, b, and c, such that they satisfy the Pythagorean theorem. This theorem states that in a right-angled triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides. In mathematical terms, it can be represented as a² + b² = c².

Characteristics of Pythagorean Triplets:
- Pythagorean triplets are unique in the sense that they form a right-angled triangle.
- The most common Pythagorean triplet is (3, 4, 5), where 3² + 4² = 5².
- Pythagorean triplets can be scaled up by multiplying all three numbers by the same integer.

Examples of Pythagorean Triplets:
- (3, 4, 5) is a Pythagorean triplet since 3² + 4² = 5² (9 + 16 = 25).
- (5, 12, 13) is another Pythagorean triplet as 5² + 12² = 13² (25 + 144 = 169).
- (8, 15, 17) forms a Pythagorean triplet with 8² + 15² = 17² (64 + 225 = 289).

Generating Pythagorean Triplets:
Pythagorean triplets can be generated using the following formulas:
- For any two positive integers m and n where m > n, the Pythagorean triplet can be expressed as a = m² - n², b = 2mn, and c = m² + n².
- By varying the values of m and n, a series of Pythagorean triplets can be obtained.
In conclusion, Pythagorean triplets are fundamental in geometry and have numerous applications in mathematics and physics. Understanding these triplets helps in solving various problems related to right-angled triangles.