Pythagorean Triangles whose Areas are Repdigit Numbers

Theorem

The following Pythagorean triangles have areas consisting of repdigit numbers:

$3-4-5$ Triangle

The triangle whose sides are of length $3$, $4$ and $5$ is a primitive Pythagorean triangle.

It has area $6$.

$693-1924-2045$ Triangle

The triangle whose sides are of length $693$, $1924$ and $2045$ is a primitive Pythagorean triangle.

It has area $666 \, 666$.

Proof

From Pythagorean Triangle whose Area is Half Perimeter, the area of the $3-4-5$ triangle is $6$, which is trivially repdigit.

The next Pythagorean triangles in area are:

the $6-8-10$ triangle, which has area $\dfrac {6 \times 8} 2 = 24$

the $5-2-13$ triangle, which has area $\dfrac {5 \times 12} 2 = 30$

So there are no more Pythagorean triangles whose areas consist of a single digit.

We have that the $693-1924-2045$ triangle is Pythagorean.

Then its area $A$ is given by:

\(\ds A\)

\(=\)

\(\ds \dfrac {693 \times 1924} 2\)

\(\ds \)

\(=\)

\(\ds \dfrac {\paren {3^2 \times 7 \times 11} \times \paren {2^2 \times 13 \times 37} } 2\)

693 and 1924

\(\ds \)

\(=\)

\(\ds \paren {2 \times 3} \times \paren {3 \times 37} \times \paren {7 \times 11 \times 13}\)

\(\ds \)

\(=\)

\(\ds 6 \times 111 \times 1001\)

6, 111 and 1001

\(\ds \)

\(=\)

\(\ds 666 \, 666\)

$\blacksquare$

Sources

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