Pythagorean triples

A Pythagorean triple is a set of three positive integers $(a, b, c)$ that satisfy $a^2 + b^2 = c^2$. They are special because they form right triangles with all-integer sides , a coincidence that has fascinated mathematicians for thousands of years.

Concept

Primitive triples. A triple is primitive if $a, b, c$ share no common factor. The smallest few are:

$(3, 4, 5), (5, 12, 13), (8, 15, 17), (7, 24, 25), (20, 21, 29), (9, 40, 41), (12, 35, 37), \dots$

Multiples. Multiplying every member of a triple by the same integer gives another triple. So $(3, 4, 5) \to (6, 8, 10) \to (9, 12, 15) \to (15, 20, 25)$, etc. Each represents the same right triangle, just at different scales.

A formula for triples. Choose any two positive integers $m > n$ with no common factor and not both odd. Then

$a = m^2 - n^2, \quad b = 2mn, \quad c = m^2 + n^2$

is always a primitive Pythagorean triple.

Examples.

Quick check. For $m=2, n=1$: $a=3, b=4, c=5$. And $9+16=25$. ✓

Why it works. Compute $a^2 + b^2$:

$(m^2 - n^2)^2 + (2mn)^2 = m^4 - 2m^2 n^2 + n^4 + 4 m^2 n^2 = m^4 + 2m^2 n^2 + n^4 = (m^2 + n^2)^2 = c^2.$

So the identity $a^2 + b^2 = c^2$ comes from a clean algebraic identity. Beautiful.

Where triples appear. Carpenters use the "$3$-$4$-$5$" rule to check that a corner is exactly $90^\circ$: measure $3$ units along one wall, $4$ along the other, and if the diagonal is $5$, the angle is square. This trick has been used in temple-building and bridge-building for thousands of years.

Worked examples

Example 1. Verify that $(7, 24, 25)$ is a triple.

• $7^2 + 24^2 = 49 + 576 = 625 = 25^2$. ✓

Example 2. Find a triple using $m = 4, n = 3$.

• $a = 16 - 9 = 7$, $b = 2 \cdot 12 = 24$, $c = 16 + 9 = 25$. (The $(7,24,25)$ triple.)

Example 3. Is $(20, 21, 29)$ a triple? Generate it with $m, n$.

• $400 + 441 = 841 = 29^2$. ✓
• $m = 5, n = 2$: $a = 25 - 4 = 21, b = 20, c = 29$. ✓

Example 4. Is $(6, 8, 10)$ primitive?

• GCD of $6, 8, 10$ is $2$, so divide by $2$: $(3, 4, 5)$ is the primitive form. So $(6,8,10)$ is not primitive.

Try it yourself

1. Verify $(8, 15, 17)$ is a Pythagorean triple.
2. Generate the triple from $m = 5, n = 4$.
3. Is $(9, 12, 15)$ a primitive triple?
4. Find the next triple after $(9, 40, 41)$ from the table (try $m = 7, n = 2$).
5. Show $(11, 60, 61)$ is a triple. With what $m, n$?
6. Carpenter's check: how would you confirm a corner is $90^\circ$ using a tape measure?
7. Generate a triple from $m = 6, n = 5$.
8. List all primitive triples with $c < 30$.

Activity / Insight

Corner check. Use a long ruler or measuring tape to verify the right angle at a corner of your classroom or kitchen. Measure $3$ ft along one wall, $4$ ft along the perpendicular wall, and check that the diagonal is exactly $5$ ft. If it is, the corner is perfectly $90^\circ$. If not, the wall is slightly skewed. This is exactly how surveyors check building corners.