Smallest Pythagorean Triangle is 3-4-5

Theorem

The smallest Pythagorean triangle has sides of length $3$, $4$ and $5$.

Proof

From Solutions of Pythagorean Equation, all Pythagorean triangles, the set of all primitive Pythagorean triples is generated by:

$\tuple {2 m n, m^2 - n^2, m^2 + n^2}$

where:

$m, n \in \Z_{>0}$ are (strictly) positive integers

$m \perp n$, that is, $m$ and $n$ are coprime

$m$ and $n$ are of opposite parity

$m > n$.

The smallest two (strictly) positive integers which satisfy the above criteria are:

$n = 1$

$m = 2$

Hence:

$2 m n = 2 \times 2 \times 1 = 4$

$m^2 - n^2 = 2^2 - 1^2 = 3$

$m^2 + n^2 = 2^2 + 1^2 = 5$

and to confirm:

$3^2 + 4^2 = 9 + 16 = 25 = 5^2$

$\blacksquare$

Sources

• 1986: David Wells: Curious and Interesting Numbers ... (previous) ... (next): $5$
• 1986: David Wells: Curious and Interesting Numbers ... (previous) ... (next): $13$
• 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $5$
• 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $13$