Triangular Triples
Problem
Three numbers a, b and c are a Pythagorean triple if $a^2+ b^2= c^2$. The triangular numbers are:
$\frac{1\times 2}{2}, \frac{2\times 3}{2}, \frac{3\times 4}{2}, \frac{4\times 5}{2}$
Show that 8778, 10296 and 13530 are three triangular numbers and that they form a Pythagorean triple.
[In fact, these are the ONLY known set of three triangular numbers that form a Pythagorean triple.]
Student Solutions
Chong Ching Tong from River Valley High School, Singapore and Andrei Lazanu, age12, School: No. 205 Bucharest, Romania approached this problem in different ways.
Here is Chong's working:
Here is Andrei's solution:
First I demonstrate that 8778, 10296 and 13530 are triangular numbers, i.e. they can be written in the form n(n+1)/2. In order to do this I decomposed the product of each of the three numbers by 2 in the hope to put it in the form n(n+1)/2. I found: $$8778\times2 = 2^2\times3\times 7\times 11\times 19 = 132\times133$$ So, $$8778 = \frac{132 \times 133}{2}$$ and so 8778 is a triangular number. $$10296\times 2 = 2^2\times 11\times13 = 143\times144$$ So, $$0296 = \frac{143 \times 144}{2}$$, and it is a triangular number. $$13530\times 2 = 2^2\times3\times5\times11\times41 = 164\times 165$$ So, $$3530 = \frac{164 \times 165}{2}$$ is also a triangular number.
Now, I demonstrate that the three numbers are a Pythagorean triple. The greatest number is 13530 $$13530^2= 183060900$$ $$8778^2+ 10296^2 = 77053284 + 106007616 = 183060900$$
So, the three numbers are a Pythagorean triple