5
$\begingroup$

i just read "mathematical constants" book;

it said that Lehmer proved the following theorem in 1900

enter image description here

where P_h(n) , P_p(n) is number of primitive Pythagorean triangles whose hypotenuses and perimeter do not exceed n respectively.

is there any simple proof for those beautiful results? Specially The first one!

$\endgroup$
0
$\begingroup$

thanks to @BarryCipra

i read the original paper.

it was AWESOME

the sketch of proof without technical details is:

consider :

$a=m^2+n^2$

$ b= m^2-n^2$

$c= 2mn$

it should be $m>n>0$ and $gcd(m,n)=1$ also $m,n$ has not the same parity.

if we want $hyp=a<N$ it must be $m^2+n^2<N$ so $m,n$ lies in a quarter of the circle with radius $\sqrt{N}$ and below the line $y=x$ whose area is equal to $$\frac{N\pi}{8}$$

Now the probability of $gcd(m,n)=1$ is equal to $\frac{6}{\pi^2}$ and the probability of both of m,n are not odd conditioned to be coprime is 2/3

So, the piece of Cake!

We Have:

$$\frac{N\pi}{8} \times \frac{6}{\pi^2} \times \frac{2}3=\frac{N}{2\pi} $$

Pyth Triangle with Hyp < N

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.