Pythagoras and Pythagorean Triples¶
The Pythagorean Theorem¶
In a right triangle with legs \(a\), \(b\) and hypotenuse \(c\):
The converse also holds: if the side lengths of a triangle satisfy this equation, the triangle is right-angled.
Example. A triangle with sides \(3, 4, 5\): \(3^2 + 4^2 = 9 + 16 = 25 = 5^2\). ✓
Pythagorean Triples¶
A Pythagorean triple is a tuple \((a, b, c)\) of natural numbers with \(a^2 + b^2 = c^2\).
Examples:
| \(a\) | \(b\) | \(c\) | Check |
|---|---|---|---|
| 3 | 4 | 5 | \(9 + 16 = 25\) |
| 5 | 12 | 13 | \(25 + 144 = 169\) |
| 8 | 15 | 17 | \(64 + 225 = 289\) |
A triple is primitive if \(\gcd(a, b, c) = 1\).
Generating All Primitive Triples¶
All primitive Pythagorean triples have the form:
with \(m > n > 0\), \(\gcd(m, n) = 1\), and \(m - n\) odd.
Example. \(m = 2, n = 1\): \(a = 4 - 1 = 3\), \(b = 4\), \(c = 4 + 1 = 5\). → \((3, 4, 5)\).
Connection to Fermat's Equation¶
The equation \(a^2 + b^2 = c^2\) has infinitely many integer solutions.
Fermat's Last Theorem states: for \(n \geq 3\), the equation
has no solution with \(a, b, c \in \mathbb{Z}^+\). The case \(n = 2\) (Pythagoras) is thus the last exponent for which integer solutions exist.
Summary¶
| Concept | Definition |
|---|---|
| Pythagorean theorem | \(a^2 + b^2 = c^2\) in a right triangle |
| Pythagorean triple | \((a,b,c) \in \mathbb{N}^3\) with \(a^2 + b^2 = c^2\) |
| Primitive triple | \(\gcd(a,b,c) = 1\) |
| Parametrization | \(a = m^2-n^2,\; b = 2mn,\; c = m^2+n^2\) |
References¶
- Hardy, G.H.; Wright, E.M.: An Introduction to the Theory of Numbers. Oxford University Press, 6th edition, 2008. Chapter 13.
- Edwards, Harold M.: Fermat's Last Theorem. Springer, 1977. Chapter 1.