Generating Pythagorean Triplets with Hypotenuse Less Than 100: A Systematic Approach

Pythagorean triplets consist of three positive integers a, b, c where a^2 b^2 c^2. While there are countless such sets, it is fascinating to find how to generate all Pythagorean triplets where the hypotenuse c is less than 100 through a systematic process.

Introduction to Pythagorean Triplets

A Pythagorean triplet is a set of three positive integers a, b, c that satisfy the equation a^2 b^2 c^2. These sets are named after the ancient Greek mathematician Pythagoras, who is credited with the theorem stating the relation between the sides of a right-angled triangle.

Systematic Generation of Triplets

There is a formula to generate all Pythagorean triplets where the hypotenuse c is less than 100. This method involves two positive integers m and n where m > n 0. The process uses the following formulas:

a m^2 - n^2 b 2mn c m^2 n^2

In order to ensure the triplet is valid, the following conditions must be met:

(m, n) 1, meaning m and n are coprime. m - n is odd, with one of them being even and the other odd. c 100.

Step-by-Step Process

To find all Pythagorean triplets with a hypotenuse less than 100, follow these steps:

Start with m 2 and incrementally increase until c exceeds 100. For each m, choose n such that n m. Ensure (m, n) 1 and m - n is odd.

Let's illustrate this process with some examples:

Example 1: For m 2:

When n 1 a 2^2 - 1^2 3 b 2 * 2 * 1 4 c 2^2 1^2 5

The triplet here is 3,4,5.

Example 2: For m 3:

When n 1 a 3^2 - 1^2 8 b 2 * 3 * 1 6 c 3^2 1^2 10

The triplet here is 6,8,10.

Complete List of Pythagorean Triplets with Hypotenuse Less Than 100

Here are some of the triplets you would generate using the above method:

3, 4, 5 5, 12, 13 6, 8, 10 7, 24, 25 8, 15, 17 9, 40, 41 12, 16, 20 12, 35, 37 14, 48, 50 20, 21, 29 28, 45, 53 30, 40, 50 36, 48, 60 39, 80, 89

Theorem on the Number of Pythagorean Triples

The number of Pythagorean triplets {a, b, n} with maximum {a, b, n} n is given by:

(frac{1}{2} prod_{p^alpha mid mid n} (2alpha - 1))

where the product is over all prime divisors p of the form 4k - 1 and the notation p^alpha mid mid n stands for p^alpha mid n and p^alpha mid n.

As an example, for n 2 cdot 3^2 cdot 5^3 cdot 7^4 cdot 11^5 cdot 13^6 the number of Pythagorean triplets with hypotenuse n is:

(frac{1}{2}(7 cdot 13 - 1) 45)

Conclusion and Further Exploration

This systematic approach to generating Pythagorean triplets with a hypotenuse less than 100 provides a structured method for understanding these sets. The underlying theorem and formulas can be complex, but they offer a deep insight into the mathematical properties of Pythagorean triplets.