Articles of pythagorean triples

Euler bricks and the $4^{th}$ dimension

I’ve heard of Euler bricks, which go like this: A Euler brick is a brick that has 3 sides, and any combination of the sides using the Pythagorean theorem will get a whole number. the Pythagorean theorem is: $$ A^2+B^2=C^2 \text{ or } \sqrt{A^2+B^2}=C$$ I am wondering… is there any 4D Euler brick? as in […]

Is there any formula to calculate the number of different Pythagorean triangle with a hypotenuse length $n$, using its prime decomposition?

Lets define $N(n)$ to be the number of different Pythagorean triangles with hypotenuse length equal to $n$. One would see that for prime number $p$, where $p=2$ or $p\equiv 3 \pmod 4$, $N(p)=0$ also $N(p^k)=0$. e.g. $N(2)=N(4)=N(8)=N(16)=0$ But for prime number $p$, where $p\equiv 1 \pmod 4$, $N(p)=1$ and $N(p^k)=k$. e.g. $N(5)=N(13)=N(17)=1$ and $N(25)=2$ and […]

Primitive Pythagorean triple generator

I was wondering how to prove the following fact about primitive Pythagorean triples: Let $(z,u,w)$ be a primitive Pythagorean triple. Then there exist relatively prime positive integers $a,b$ of different parity such that $$z = a^2-b^2, \quad u = 2ab, \quad \text{and} \quad w = a^2+b^2.$$ I see how $z,u,w$ must form the sides of […]

Solving Pythagoras Problem

An aircraft hangar is semi-cylindrical, with diameter 40m and length 50 m. A helicopter places an inelastic rope across the top of the hangar and one end is pinned to a corner, a A. The rope is then pulled tight and pinned at the opposite corner, B. Determine the lenghth of the rope. So, first […]

Proof that the sum of the even side and the hypotenuse of a coprime (and positive) Pythagorean triple is a square number

This question already has an answer here: Formulas for calculating pythagorean triples 7 answers

Is there a way to see this geometrically?

In my answer to this question – Finding the no. of possible right angled triangle. – I derived this result: If a right triangle has integer sides $a, b, c$ and integer inradius $r$, then all possible values of $a$ and $b$ can be gotten in terms of $r$ as follows: For every possible divisor […]

Mohanty's conjecture

Is there any proof or counter-proof of Mohanty’s conjecture (1988) in the litterature: The numbers n, n + 6, and n + 12 cannot be expressed simultaneously as sum of two squares.

Find all solutions to the equation $x^2 + 3y^2 = z^2$

Find all positive integer solutions to the equation $x^2 + 3y^2 = z^2$ So here’s what I’ve done thus far: I know that if a solution exists, then there’s a solution where (x,y,z) = 1, because if there is one where $(x,y,z) = d$, then $\frac{x}{d}, \frac{y}{d}, \frac{z}{d}$. is also a solution. I’m trying to […]

Find pairs of side integers for a given hypothenuse number so it is Pythagorean Triple

I am trying to find all pairs of side integers (a, b) for a given hypothenuse number n so that (a, b, n) is a Pythagorean triple, i.e.,$ a^2 + b^2 = n^2$ The approach i am using is Sorting the array in ascending order Finding the square of each element in array for(j->0 to […]

solutions of $a^2+b^2=c^2$

I am trying to figure the following out. If you have $a^2+b^2=c^2$ and let $x=a/c$ and $y=b/c$ how can you show that $x=\frac{m^2-n^2}{m^2+n^2}$ and $y=\frac{2mn}{m^2+n^2}$ for some relatively prime numbers $m,n \in \mathbb{Z}$