---

Problem 75

---

Singular integer right triangles

It turns out that $12$ 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.

$12$ cm: $(3,4,5)$

$24$ cm: $(6,8,10)$

$30$ cm: $(5,12,13)$

$36$ cm: $(9,12,15)$

$40$ cm: $(8,15,17)$

$48$ cm: $(12,16,20)$

In contrast, some lengths of wire, like $20$ 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 $120$ cm it is possible to form exactly three different integer sided right angle triangles.

$120$ 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?

---

唯一的整数边直角三角形

若电线只能以唯一方式弯折成整数边直角三角形，则电线的最短长度是$12$厘米；当然，还有很多种长度的电线都只能以唯一方式弯折成整数边直角三角形，例如：

$12$厘米: $(3,4,5)$

$24$厘米: $(6,8,10)$

$30$厘米: $(5,12,13)$

$36$厘米: $(9,12,15)$

$40$厘米: $(8,15,17)$

$48$厘米: $(12,16,20)$

相反地，有些长度的电线，比如$20$厘米，不可能弯折成任何整数边直角三角形，而另一些长度的电线则有多种解法；例如，$120$厘米的电线可以弯折成三个不同的整数边直角三角形。

$120$厘米: $(30,40,50), (20,48,52), (24,45,51)$

记电线长度为$L$，对于$L \le 1,500,000$，有多少种电线长度只能以唯一方式弯折成整数边直角三角形？

---