Problem Archive

It turns out that $\pu{12 cm}$ is the smallest length of wire that can be bent to form an integer sided right angle triangle in exactly one way, but there are many more examples.

• $\pu{\mathbf{12} \mathbf{cm}}$: $(3,4,5)$
• $\pu{\mathbf{24} \mathbf{cm}}$: $(6,8,10)$
• $\pu{\mathbf{30} \mathbf{cm}}$: $(5,12,13)$
• $\pu{\mathbf{36} \mathbf{cm}}$: $(9,12,15)$
• $\pu{\mathbf{40} \mathbf{cm}}$: $(8,15,17)$
• $\pu{\mathbf{48} \mathbf{cm}}$: $(12,16,20)$

In contrast, some lengths of wire, like $\pu{20 cm}$, cannot be bent to form an integer sided right angle triangle, and other lengths allow more than one solution to be found; for example, using $\pu{120 cm}$ it is possible to form exactly three different integer sided right angle triangles.

• $\pu{\mathbf{120} \mathbf{cm}}$: $(30,40,50)$, $(20,48,52)$, $(24,45,51)$

Given that $L$ is the length of the wire, for how many values of $L \le 1\,500\,000$ can exactly one integer sided right angle triangle be formed?