Pythagorean Triangles whose Areas are Repdigit Numbers
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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): $13$