Counting Right Triangles

Calculus Level 5

$P(n)$ and $H(n)$ respectively denotes the number of primitive right angled triangles whose perimeters, hypotenuses do not exceed $n$ .

Find $\displaystyle\lim_{n\to\infty}\dfrac{P(n)}{H(n)}$

\dfrac{\ln3}{\pi}

\dfrac{\ln2}{\pi}

\dfrac{\ln4}{\pi}

\dfrac{\ln2}{2\pi}

1 solution

Mark Hennings
Feb 2, 2019

Results by Lehmer from the start of the 20th century tell us that $H(n) \; \sim \; \frac{n}{2\pi} \hspace{2cm} P(n) \; \sim \; \frac{n \ln 2}{\pi^2} \hspace{2cm} n \to \infty$ and hence $\lim_{n \to \infty}\frac{P(n)}{H(n)} \; = \; \frac{2\ln 2}{\pi} \; = \; \boxed{\frac{\ln4}{\pi}}$

Is the derivation of these results at all palatable? I had to resort to coding, but I'd be interested to know if there were any approachable way of getting to the answer. (I wonder if the form of the ratio simplifies things at all? Or the fact that both $P(n)$ and $H(n)$ are roughly linear functions of $n$ ?)

Chris Lewis - 2 years, 4 months ago

If you chase down the references in Mathworld and elsewhere you find Lehmer's 1900 article , which is available on Jstor.

Mark Hennings - 2 years, 4 months ago

Counting Right Triangles

1 solution

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