Pythagorean Triples

Statement

A Pythagorean triple is a triple of positive integers $(a, b, c)$ satisfying

Every primitive Pythagorean triple (with $\gcd(a,b,c) = 1$ and $b$ even) has the form

where $m > n > 0$, $\gcd(m, n) = 1$, and $m \not\equiv n \pmod{2}$.

Every Pythagorean triple is obtained by multiplying a primitive one by a common factor $k \geq 1$.

Visualization

Primitive triples generated by small $(m, n)$:

m   n   a = m²-n²   b = 2mn   c = m²+n²   Check
----+---+-----------+---------+-----------+------
2   1       3           4         5        3²+4²=25=5²   ✓
3   2       5          12        13        5²+12²=169=13² ✓
4   1      15           8        17        15²+8²=289=17² ✓
4   3       7          24        25        7²+24²=625=25² ✓
5   2      21          20        29        21²+20²=841=29²✓
5   4       9          40        41        9²+40²=1681=41²✓

The first triple $(3, 4, 5)$ is the simplest and best known. All others in the table above are primitive because $\gcd(m,n) = 1$ and $m \not\equiv n \pmod 2$.

Proof Sketch

1. Odd/even structure: In a primitive triple, exactly one of $a, b$ is even. Say $b = 2s$ is even. Then $a^2 = c^2 - b^2 = (c-b)(c+b)$, and one checks $\gcd(c-b, c+b) = 2$.

2. Unique factorization: Write $c - b = 2n^2$ and $c + b = 2m^2$ (using that the product of two coprime numbers each equalling a perfect square forces each factor to be a square). Then $c = m^2 + n^2$ and $b = 2mn$.

3. Complete classification: Conversely, for any $m > n > 0$ with $\gcd(m,n) = 1$ and $m \not\equiv n \pmod 2$, the triple $(m^2 - n^2, 2mn, m^2 + n^2)$ is primitive.

4. Every triple: Multiply any primitive triple by $k \in \mathbb{Z}_{>0}$ to get $(ka, kb, kc)$.

Connections

The proof uses unique factorization from the Fundamental Theorem of ArithmeticFundamental Theorem of Arithmetic$$n = p_1^{a_1} \cdot p_2^{a_2} \cdots p_k^{a_k}$$Every integer greater than 1 has a unique prime factorizationRead more →. The structure of $\mathbb{Z}[i]$ (Gaussian integers) provides a slick alternative proof — each factor $c + ib$ or $c - ib$ is a product of Gaussian primes, tying this to Sum of Two SquaresSum of Two Squares$$p = a^2 + b^2 \iff p = 2 \text{ or } p \equiv 1 \pmod{4}$$A prime is a sum of two perfect squares if and only if it equals 2 or is congruent to 1 mod 4Read more →. Pythagorean triples also connect to Euler's Totient (counting coprime pairs) and to Diophantine equations more broadly, which uses ideas from Chinese Remainder TheoremChinese Remainder Theorem$$\mathbb{Z}/mn\mathbb{Z} \cong \mathbb{Z}/m\mathbb{Z} \times \mathbb{Z}/n\mathbb{Z}$$Simultaneous congruences with coprime moduli have a unique solution mod their productRead more →.

/-- Mathlib defines `PythagoreanTriple x y z` as `x * x + y * y = z * z` (over ℤ).
    The triple (3, 4, 5) satisfies this by a direct numerical check. -/
theorem pyth_345 : PythagoreanTriple 3 4 5 := by
  unfold PythagoreanTriple
  norm_num

/-- Scaling: if (a, b, c) is a Pythagorean triple, so is (k·a, k·b, k·c). -/
theorem pyth_scaled (x y z k : ℤ) (h : PythagoreanTriple x y z) :
    PythagoreanTriple (k * x) (k * y) (k * z) :=
  h.mul k