# 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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(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 $\... 2 votes 1 answer 117 views ### Is the angle of the 345 triangle (pythagorean triple) related to the Geometric Progression. 4, 2, 1, 1/2, 1/4? I realised recently I could use this equation for the pythagorean triples. As you can see the 345 triangle seems to be special? as we can create it using x=2 AND x=3. I know the 345 triangle is ... 5 votes 3 answers 149 views ### Is the set$\ \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]\ $and how do you show this? It seems likely true based on ... 0 votes 2 answers 71 views ### Hypotenuses for which there exist exactly 4 distinct integer triangles with an extra constraint A084648 of the OEIS contains all numbers where the square of the number can be decomposed exactly in four different ways in a sum of two squares of integers. For example 65 is a term of A084648 ... 2 votes 2 answers 141 views ### Three circles internaly tangent to an equilateral triangle The diagram shows an equilateral triangle of side length 1 with 3 identical circles. Find the radius of the circle. The correct answer for the length of the right triangle in red should be$\sqrt{3}r$... 2 votes 1 answer 73 views ### Generating Pythagorean triples$(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 ... 9 votes 2 answers 198 views ### 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 ...
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 $$\begin{array}{|c|c|c|} \hline A² &B²&C² \\ \hline D²&E²&F²\\... 2 votes 1 answer 66 views ### Square pyramid with integer measurements. A right square pyramid has the following dimensions: Length of the base square: 2a Height (distance from the vertex to the base center): h Distance from the vertex to the middle point of the base ... 0 votes 0 answers 34 views ### Color each positive integer with red and blue, whether there must be three numbers a, b, c with same color such that a^2+b^2 = c^2? [duplicate] Color each positive integer with red and blue, whether there must be three numbers a, b, c with same color such that a^2+b^2 = c^2? Such as if 3,4,5 is blue, then 3^2 + 4^2 = 5^2. Is there a ... -1 votes 2 answers 61 views ### If A^2+B^2=C^2 with A odd, A,B,C coprime and A<B<C, is B+C a square? [closed] I was looking through Pythagorean triplets and noticed something: Take 3 numbers A,B,C such that: A^2+ B^2= C^2 A<B<C A,B and C have no common factors A is odd Prove that B+C ... 0 votes 1 answer 85 views ### Primitive Pythagorean Triples + Exponent So, a while ago, I watched a YouTube video about the positive integer solution of$$3^x + 4^y = 5^z$$and the result was  x = y = z = 2 . My question, now, is: For any primitive Pythagorean ... 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 ... 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^\... 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 ... 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: ... 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 ... 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 ... 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+... 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 ... 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 ... 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^... 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 "... 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 ... 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 ... 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, ... 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 ... 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... 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 ... 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, ... -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 ... 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 ... 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 ... 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, ... 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} \... 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. (... 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 ... 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 ... 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 \... 