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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9 votes
2 answers
190 views

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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7 votes
2 answers
121 views

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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1 vote
1 answer
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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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1 vote
1 answer
42 views

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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3 votes
2 answers
112 views

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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2 votes
1 answer
97 views

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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1 vote
1 answer
88 views

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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1 vote
2 answers
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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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2 votes
1 answer
121 views

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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2 votes
1 answer
117 views

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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5 votes
3 answers
149 views

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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0 votes
2 answers
71 views

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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2 votes
2 answers
141 views

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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  • 549
2 votes
1 answer
73 views

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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9 votes
2 answers
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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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4 votes
2 answers
176 views

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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2 votes
1 answer
66 views

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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0 votes
0 answers
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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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0 votes
1 answer
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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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3 votes
4 answers
306 views

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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  • 61
1 vote
2 answers
45 views

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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3 votes
1 answer
102 views

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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4 votes
3 answers
126 views

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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  • 307
2 votes
1 answer
29 views

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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30 votes
5 answers
1k views

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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2 votes
2 answers
178 views

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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7 votes
1 answer
136 views

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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4 votes
0 answers
89 views

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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  • 696
2 votes
1 answer
178 views

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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  • 571
2 votes
3 answers
127 views

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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1 vote
1 answer
109 views

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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2 votes
1 answer
193 views

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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16 votes
6 answers
459 views

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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0 votes
1 answer
59 views

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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2 votes
2 answers
88 views

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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  • 3,399
3 votes
1 answer
280 views

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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3 votes
4 answers
217 views

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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-1 votes
1 answer
235 views

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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4 votes
4 answers
180 views

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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  • 359
3 votes
1 answer
63 views

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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4 votes
4 answers
275 views

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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  • 649
3 votes
2 answers
166 views

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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  • 549
0 votes
1 answer
66 views

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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1 vote
2 answers
90 views

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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  • 1,559
0 votes
0 answers
31 views

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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2 votes
1 answer
74 views

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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0 votes
1 answer
100 views

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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