Questions tagged [pythagorean-triples]
Questions about Pythagorean triples, positive integer solutions to $a^2 + b^2 = c^2$.
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Generating Pythagorean triples using reduced fractions
I know that this question has answers here, but I'm looking for a way to make the following argument work.
Let $(a,b,c)$ be a primitive Pythagorean triple with $a$ even. The goal is to prove that we ...
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Connection between Laplace Transforms and Pythagorean triples
I was studying for a recent university exam when I realized that there appears to be a connection between the Laplace transforms of certain functions and Pythagorean triples.
Mainly the Laplace of $t\...
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Generating Pythagorean triples where the legs are Hypotenuses of other Pythagorean triples
I know how to generate regular Pythagorean Triples given two positive integers P and Q such that
$$a=2*p*q$$ $$b=p^2-q^2$$ $$c=p^2+q^2$$ where $p>q$, but I want to find scenarios where $a$ and $b$ ...
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Why is leg of the pythagorean triangle which is of the form qp^3 or 16p where q & p are odd has exactly 10 solutions?
The question is in the name. So for example the number 48 which is of the form $q * p^3$ can be the leg of exactly 10 Pythagorean triangles. Why is that? I saw it in a text on number theory and I've ...
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how would I find a Pythagorean triple that is all square numbers? (a^2)^2 + (b^2)^2 = (c^2)^2 [duplicate]
I started out with the thought that I could use the imaginary plain to help me as you use that to find Pythagorean triples however I have gotten nowhere can anyone help?
at the end I should have a ...
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Tangential Equidiagonal (Irregular) Quadrilateral with integer coordinates
Playing with mathematics this week-end, I ended up with this question I couldn't solve, so I'm asking here.
I try to find a Quadrilateral ABCD with integers (cartesian) coordinates (points are on a ...
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Pythagorean triples and primes. [duplicate]
Determine whether there are any right-angled triangles with integer lengths such that the
lengths of both of the sides adjacent to the right angle are primes.
This was the question I was asked by my ...
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Does the proof for primitive Pythagorean triples follow that the formula, $x=q^2-p^2, y=2pq, z=p^2+q^2,$ give all primitive Pythagorean triples?
Essentially, I am aware of the proof that if a primitive Pythagorean triple is of the form $x^2+y^2=z^2,$ then $x=q^2-p^2,$ $y=2pq$ and $z=p^2+q^2.$ But I'm not going to write this proof here.
My ...
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Criterion for being the length of a hypotenuse
I'm wondering if the criterion I give below can be simplified. Its goal is to see whether some integer $n\geq 1$ is the length of the hypotenuse of some right-angled triangle with integer side lengths....
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How we may express four squares whose difference is each a square in terms of (preferably solid) geometry?
The problem of finding four squares whose difference is each a square is much more exhaustive as I thought. A quest up to $2^{34}$ yields nothing. The largest almost solution found in the range up to $...
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What is the equation (or equations) to find the sequence/s of odd numbers that are the result of the sum of two squares?
(not including $0^2 + b^2$ and only including odd numbers:)
Starting at $5 = 1^2 + 2^2$
then
$13 = 2^2 + 3^2$
$17 = 4^2 + 1^2$
$25 = 3^2 + 4^2$
$29 = 2^2 + 5^2$
What is the equation/s to find the
$\...
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Is the angle of the 345 triangle (pythagorean triple) related to the Geometric Progression. 4, 2, 1, 1/2, 1/4?
I realised recently I could use this equation for the pythagorean triples.
As you can see the 345 triangle seems to be special? as we can create it using x=2 AND x=3.
I know the 345 triangle is ...
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Is the set $\ \left\{ \frac{b-a }{c-a}:\ (a,b,c)\ \text{is a primitive Phythagorean triple with}\ a<b<c\ \right\}\ $ dense in $\ [0,1]\ ?$
Is the set $\ \left\{ \frac{b-a }{c-a}:\ (a,b,c)\ \text{is a primitive Pythagorean triple with}\ a<b<c\ \right\}\ $ dense in $\ [0,1]\ $ and how do you show this?
It seems likely true based on ...
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Hypotenuses for which there exist exactly 4 distinct integer triangles with an extra constraint
A084648 of the OEIS contains all numbers where the square of the number can be decomposed exactly in four different ways in a sum of two squares of integers. For example 65 is a term of A084648 ...
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Three circles internaly tangent to an equilateral triangle
The diagram shows an equilateral triangle of side length 1 with 3 identical circles. Find the radius of the circle.
The correct answer for the length of the right triangle in red should be $\sqrt{3}r$...
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Generating Pythagorean triples $(a,b,c)$ such that $b>a+n$ for integer $n$, and $a+b$ is minimum
I'm trying to generate a Pythagorean triple, that is, $a,b,c \in \mathbb N$ that
$$a^2 + b^2 = c^2$$
under the condition that $b > a + n$, $n \in \mathbb N$ and $a+b$ is minimum.
I tried expanding ...
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For integers $x<y<z$, why are these cases impossible for Mengoli's Six-Square Problem?
I got inspired by this question "Four squares such that the difference of any two is a square?" and rewrote zwim's program that is provided by his answer to the question "Solutions to a ...
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Magic square of square
I am trying to solve magic square of square
First let me explain my approach when will be the magic square form if we have
$$\begin{array}{|c|c|c|} \hline A² &B²&C² \\ \hline D²&E²&F²\\...
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Square pyramid with integer measurements.
A right square pyramid has the following dimensions:
Length of the base square: $2a$
Height (distance from the vertex to the base center): $h$
Distance from the vertex to the middle point of the base ...
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Color each positive integer with red and blue, whether there must be three numbers $a, b, c$ with same color such that $a^2+b^2 = c^2$? [duplicate]
Color each positive integer with red and blue, whether there must be three numbers $a, b, c$ with same color such that $a^2+b^2 = c^2$?
Such as if $3,4,5$ is blue, then $3^2 + 4^2 = 5^2$. Is there a ...
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If $A^2+B^2=C^2$ with $A$ odd, $A,B,C$ coprime and $A<B<C$, is $B+C$ a square? [closed]
I was looking through Pythagorean triplets and noticed something:
Take $3$ numbers $A,B,C$ such that:
$A^2+ B^2= C^2$
$A<B<C$
$A,B$ and $C$ have no common factors
$A$ is odd
Prove that $B+C$ ...
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Primitive Pythagorean Triples + Exponent
So, a while ago, I watched a YouTube video about the positive integer solution of $$3^x + 4^y = 5^z$$
and the result was $ x = y = z = 2 $.
My question, now, is:
For any primitive Pythagorean ...
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How to calculate the side of a right triangle from the coordinates of points and the length of one side?
I have the line AB. And I need to calculate the coordinates of point D.
I know the coordinates of points A, B and C.
If I make this an imaginary right triangle, I just need to know the length of the ...
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How can I calculate the relative lengths of triangle sides if all angles are known?
I've read about this on other occasions on here before but I think my problem isn't a duplicate.
I'm trying to find the lengths of sides of a triangle where I know all three angles.
Let's say $A = 60^\...
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Upper and lower bound on the N-th Pythagorean triplet
Let $H_n$ be the hypotenuse of the $n$-th primitive Pythagorean Triplet when arranged in ascending order of the length of the hypotenuse. What is known about the asymptotic expansion of or bounds on $...
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Given two pythagorean triples, generate another
I don't know if this has been asked before, but I could not find any existing answer.
I noticed that for any pair of primitive pythagorean triples (not necessarily distinct), let's say:
...
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Does the edge of the shadow of a person's head move in a straight line if the person moves in a straight line with the light source fixed in a place?
So, let's say that there is a light source at a height "H" above the ground. A person of height "x" starts moving in a straight line with uniform velocity (not direct under the ...
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How do you find Pythagorean triples that approximately correspond to a right triangle with a given angle?
Given an angle $\theta$, can I find a Pythagorean triple $(A,B,C)$ such that the corresponding right triangle contains an angle that is as close to $\theta$ as I want? And if so, how? For example ...
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Does the diophantine equation $ax^2+by^2=cz^2+d$ always have solutions?
Let $a,b,c,d,x,y,$ and $z \in \mathbb{N}$ where $a,b,c,$ and $d$ are constants but d is allowed to be zero .
$ax^2+by^2=cz^2+d$
First example :
when $a=b=c=1 $ and $d=0$ we have the equation : $x^2+...
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Pythagoras Theorem or Trigonometry?
I had this question in my test:-
Authority wants to construct a slide in a city park for children. Authority prefers the top of the slide at a height of 4m above the ground and inclined at an angle of ...
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Enumerating a subset of rationals: a variation on the Calkin–Wilf tree
I've been doing some recreational maths on the Calkin–Wilf tree, which enumerates all positive irreducible fractions in the form of a binary tree, similar to the Stern–Brocot tree. Instead of ...
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A Function to Generate Pythagorean Triples
For a given right triangle $ABC$ with hypotenuse $C$, if you know that value of one of the legs, such as $B$, you can calculate possible values for the other two sides by using the equation
$"\dfrac{B^...
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Weighted Pythagorean Triples [duplicate]
Let $r,s,t$ be positive integers. Do there exist integers $a,b,c$, not all 0, satisfying $$r a^2+s b^2=t c^2\enspace ?$$
Let's call $(r,s,t)$ "valid" if such a solution exists, and "...
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A "simple" proof about triangles and relativity (similar triangles and Pythagoras) [closed]
In "ABC of relativity" Bertrand Russel claims that the following is easy to prove but I'm stuck.
In the image $OD=OC$ and $OY=OX$, he claims that $OC^2-OX^2=SZ^2-RQ^2$
Also every angle that ...
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Why is $\big|\big((2n-1)^2-2k^2\big)\big|$ always a specific prime product?
One would think that $\big((2n-1)^2-2k^2\big)$ could be any odd number but it is always
$\big|(2n-1)^2-2k^2\big|
\in\big\{1, 7,17,23,31,41,\cdots\big\}
, (n,k)\in\mathbb{N}, GCD\big((2n-1),k\big)=1$
...
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Near-Pythagorean triplets: What are the general solutions to $a^2+b^2=c^2-1$?
Obtaining the most general solution to a quadratic Diophantine equation in three variables is often easier if the equation is homogeneous. For example, by focusing on "primitive" solutions, ...
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absolute value and pythag theorem
The positive difference between real numbers c and d are defined to be absolute values of c minus d. The hypotenuse of the right triangle is 5 cm and the perimeter is 11 cm. Find the number of cm in ...
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Unusual primitive Pythagorean triple identity
I am working on another project relating to Pythagorean triples and came across an unusual property
The primitive triple generator $\left(n, \frac{n^2-1}{2}, \frac{n^2+1}{2}\right)$ for $n\in\mathbb N$...
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How is the Pythagorean triple ternary tree constructed?
I don't understand the structure of the ternary tree of Pythagorean triples. I can see that each triple to the left is linked to larger triples on the right but I can see no pattern in how the triples ...
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What’s the most natural/useful way of ranking Pythagorean triples “by size”?
I want to “rank” [primitive] Pythagorean triples by some metric that could reasonably be referred to as “size”.
Naturally, there are a huge number of options: size of hypotenuse, size of smallest leg, ...
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Rational Solution to $\frac{x-x^3}{(x^2+1)^2} + \frac{y-y^3}{(y^2+1)^2} = \frac{z-z^3}{(z^2+1)^2}$
Find a (all) rational solutions, 0<(x,y,z)<1 to the equation:
$$\frac{x-x^3}{(x^2+1)^2} + \frac{y-y^3}{(y^2+1)^2} = \frac{z-z^3}{(z^2+1)^2}$$
or else show there are no rational solutions.
This ...
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What is my mistake in finding this pythagorean triplet? [closed]
Since Project Euler copyright license requires that you attribute the problem to them, I'd like to add that this is about question 9 there.
I am trying to solve this problem on only two brain cells ...
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Side length of the squares
Consider the following figure where $ORQP$ and $RSTQ$ are squares of same side length. $O,R,T$ lies on the circle with centre $B$.The radius of the circle is $5$ units. Find the side length of the ...
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Difference between hypotenuse and larger leg in a Pythagorean triple
I've been number crunching irreducible Pythagorean triples and this pattern came up: the difference between the hypotenuse and the larger leg seems to always be n² or 2n² for some integer n. Moreover, ...
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A discussion about exponential diophantine equations and pythagorean triples
Do all pythagorean triples $(a,b,c)$ have the identity that the three
(exponential diophantine) equations
\begin{equation} a^x+b^y=c^z \end{equation} \begin{equation}
b^y+c^z=a^x \end{equation} \...
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A different Pythagorean Theorem. [duplicate]
I have been finding a lot of numbers that satisfy the equation a^2 +b^2 = c^2 +d^2, and I was wondering if there was a way to generate a, b, c, and d, using 2 or 3 starting numbers like m or n. (...
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Given distance between cars, how to calculate the relative position and identify if the car is on my left or right?
I am doing decentralized control of vehicles. Assume I am a car with $2$ sensors, sensor $A$ in front of me and sensor $B$ behind. Using these sensors I can measure the distance from me to other ...
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Distance between 2 points and time taken - Speed keeps increasing as the distance get further why?
I am trying to calculate the time taken between 2 points for a game with X and Y coordinates.
When I plot in the actual time taken based on the game data, and the X Y coordinates, I can't find a ...
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Vector Geo Problem I am lost in.
Question:
There exists positive integers $x, y, z$ where $\gcd(x,y,z)=1$, so that for any Pythagorean triple $(a,b,c)$
$$\begin{pmatrix} x & y & y \\ y & x & y \\ y & y & z \...
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Does it follow from the Pythagorean theorem that "there is no right-angled triangle with sides 5, 10 and 11"?
Assume, for the sake of this question, that we define the Pythagoeran theorem as
Theorem 1: In a right-angled triangle, the sum of the squares of the two shortest sides is equal to the square of the ...
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