Question

Three integers \((x, y, z) \in \mathbb{N}^{3}\) form a Pythagorean triple if \(x^{2}+y^{2}=z^{2}\), and a primitive Py. thagorean triple if furthermore gcd \((x, y, z)=1\). (i) Show that any Pythagorean triple is of the form \((\lambda x, \lambda y, \lambda z)\) for a primitive Pythagorean triple \((x, y, z)\) and some \(\lambda \in \mathbb{N}\), (ii) Let \(s, t \in N\) be coprime with \(s>t\) and \(s t\) even. Show that \(\left(s^{2}-t^{2}, 2 s t, s^{2}+t^{2}\right)\) is a primitive Pythagorean triple, (iii) Let \((x, y, z)\) be a primitive Pythagorean triple. Show that \(z\) is odd and either \(x\) or \(y\) is odd, but not both. Hint: Calculate modulo 4. Assume that \(x\) is odd. Prove that \((z+x) / 2\) and \((z-x) / 2\) are coprime squares, not both odd, and conclude that \((x, y, z)\) is of the form as in (ii). (iv) Use (ii) and (iii) to generate all primitive Pythagorean triples \((x, y, z)\) with \(z \leq 100\).

Short answer

Any Pythagorean triple can be written from primitives with scaling; use coprimes \((s, t)\) to form these primitives.

Step-by-step solution

Understanding Pythagorean Triples

A Pythagorean triple consists of integers \((x, y, z)\) such that \(x^2 + y^2 = z^2\). For it to be primitive, \(\gcd(x, y, z) = 1\). This means that the triple shares no common divisors other than 1. The first step is to understand the form of these triples.

Identifying the Primitive Form

For any Pythagorean triple, we can express it as \((\lambda x, \lambda y, \lambda z)\), where \((x, y, z)\) is a primitive Pythagorean triple, and \(\lambda\) is a positive integer. This is because we can factor out the greatest common divisor (\(\lambda\)) from the triple, reducing it to a primitive form.

Using Coprimes to Form Primitive Pythagorean Triples

Consider two coprime integers \(s\) and \(t\) with \(s > t\) and \(st\) even. The triple \((s^2 - t^2, 2st, s^2 + t^2)\) solves the equation \(x^2 + y^2 = z^2\) for these variables. Since \(s\) and \(t\) are coprime, the resulting triple is primitive.

Analyzing Modulo 4

In modulo 4 arithmetic, perfect squares are congruent to either 0 or 1. For \(x^2 + y^2 = z^2\), since it's assumed \(x\) is odd, observe that \(x^2 \equiv 1 \mod 4\). Therefore, since \(y^2 \equiv 1 \mod 4\) is impossible (\(2 \equiv 2 \mod 4\)), \(z\) must be odd (\(\equiv 1 \mod 4\)) and \(x\) or \(y\) must be even (\(\equiv 0 \mod 4\)).

Proving Coprime Squares

Set \((z+x)/2 = a^2\) and \((z-x)/2 = b^2\) such that \((a^2 - b^2, 2ab, a^2 + b^2)\) forms a primitive Pythagorean triple satisfying \((x, y, z)\). If \((z+x)/2\) and \((z-x)/2\) are coprime squares, this leads to \((x, y, z)\) being representable in this unique way.

Generating All Primitives with Constraints

To generate all primitive Pythagorean triples with \(z \leq 100\), vary \(s > t\) while ensuring \(st\) is even. For each pair, compute \(s^2 - t^2\), \(2st\), and \(s^2 + t^2\). Ensure all solutions have values for \(z\) less than or equal to 100.

Key concepts

Primitive Pythagorean Triples

A primitive Pythagorean triple consists of three integers \((x, y, z)\) such that they satisfy the equation \(x^2 + y^2 = z^2\) and have the additional property that their greatest common divisor (GCD) is 1. This means there are no other common divisors of \(x\), \(y\), and \(z\) except for the number 1.

Primitive triples represent the "pure" form of Pythagorean triples. Any other Pythagorean triple can be derived from them. Here's how it works:

• Suppose \((x, y, z)\) is a primitive Pythagorean triple.
• We can multiply each component of this triple by a positive integer \(\lambda\) to form another Pythagorean triple \((\lambda x, \lambda y, \lambda z)\).
• This process does not affect the Pythagorean nature (since \((\lambda x)^2 + (\lambda y)^2 = (\lambda z)^2\) still holds true).
• However, it results in a non-primitive form as the new triple could have a greater GCD (usually \(\lambda\)).

Understanding this concept helps in identifying the foundational Pythagorean triples from which others can be constructed.

Coprime Integers

Coprime integers are a pair or set of numbers whose greatest common divisor (GCD) is 1. This means they have no shared prime factors. Why does this matter for Pythagorean triples? Let's explore:

To construct a primitive Pythagorean triple, use coprime numbers. Here’s a quick guide:

• Choose integers \(s\) and \(t\) such that \(s > t\).
• Another rule is \(st\) should be even (which implies one of the numbers is even).
• Once you have \(s\) and \(t\), form the triple using the following expressions: \((s^2 - t^2, 2st, s^2 + t^2)\).
• This method ensures that the triple remains primitive if \(s\) and \(t\) are coprime.

By starting with coprime integers, it guarantees that the resulting Pythagorean triple does not share any common divisor other than 1. This offers a powerful method for generating primitive triples systematically.

Modulo Arithmetic

Modulo arithmetic, sometimes called clock arithmetic, is a system of arithmetic for integers. It considers the remainder when one number is divided by another. In Pythagorean triples, modulo arithmetic helps us analyze the properties of these integers.

Let's consider why modulo 4 arithmetic is especially useful:

• We know squares take the form \(0\) or \(1\) when considered modulo 4. For instance, \(x^2 \equiv 1 \mod 4\) if \(x\) is odd.
• This means that if \(x^2 + y^2 = z^2\), and \(x\) is odd, \(x^2 \equiv 1\) and forces \(z\) to also be odd (since \(1 + \text{even} eq \text{odd}\)).
• Moreover, if one of \(x\) and \(y\) is odd, the other must be even to satisfy the equation.

Modulo arithmetic clearly illustrates that in a primitive Pythagorean triple, \(z\) has to be odd, and one of \(x\) or \(y\) must be even. This insight is crucial for deducing the forms and constraints of these meaningful sets of numbers.