Section 1.1: Method 1 - Rearranging the Formula

Assume that (x, y, z) forms a Pythagorean Triple. We can restate the equation x² + y² = z² as:

x² = z² - y² = (z - y)(z + y)

From this, we derive the relationship:

x / (z - y) = (z + y) / x

Note that division by zero is not a concern here, as if z = y, it would imply x = 0, which is not permissible for a Pythagorean Triple.

Since x, y, and z are all positive integers, the expression x / (z - y) must represent a rational number. A rational number is defined as any number that can be expressed as m/n, where m and n are integers and n ≠ 0. For example, 0.5 is rational as it equals 1/2.

We can equate:

x / (z - y) = (z + y) / x = m/n

This leads us to these two equations:

1. (z - y) / x = n/m (the reciprocal of the first)
2. (z + y) / x = m/n

By adding these equations, we derive:

2z / x = n/m + m/n = (n² + m²) / mn

This allows us to define:

x = 2mn and z = n² + m².

By subtracting the second equation from the first, we find:

2y / x = m/n - n/m = (m² - n²) / mn,

which gives us:

y = m² - n².

Thus, we have shown that any Pythagorean Triple can be represented as:

• x = 2mn
• y = m² - n²
• z = m² + n²

It is important to note that x² + y² = (2mn)² + (m² - n²)² = z², confirming that any values of m and n (with m > n > 0) will yield a valid Pythagorean Triple.

The first video titled Euclid's Pythagorean Triples further elucidates the process of deriving Pythagorean Triples using Euclid's Formula. It offers visual explanations and examples that enhance understanding.

Section 1.2: Method 2 - Primitive Pythagorean Triples

Another approach to deriving Euclid’s Formula involves examining primitive Pythagorean Triples, where x, y, and z share no common factor. For example, (3, 4, 5) is primitive, while (6, 8, 10) is not.

Here are some important lemmas to consider:

Lemma A: z is odd

If z were even, it could be expressed as z = 2Z for some integer Z, leading to x² + y² = (2Z)² = 4Z². This implies that 4 divides x² + y². However, a square can only be divisible by 4 or leave a remainder of 1 when divided by 4. Thus, both x and y must also be even, contradicting the assumption that (x, y, z) is primitive.

Lemma B: Either x or y is even, but not both

This lemma can be reasoned similarly. Since z is odd, z² is odd, which requires that either x² or y² must be even (but not both) to maintain the integrity of the equation.

To derive Euclid's Formula, let's assume x is even. This leads to:

x² = z² - y² = (z - y)(z + y)

Both z - y and z + y are even. By introducing variables a, b, and c, we can express our findings as follows:

• x = 2a
• y = 2b
• z = b + c

Additionally, since both b and c must be coprime, we conclude that:

b = m² and c = n², resulting in the final expressions:

• x = 2mn
• y = m² - n²
• z = m² + n²

Thus, we have confirmed that any primitive Pythagorean Triple can be expressed in the form provided by Euclid's Formula.

The second video titled Formula for Generating ALL Pythagorean Triples expands on the concepts discussed, illustrating the derivation and applications of Pythagorean Triples in various contexts.