# Questions tagged [pythagorean-triples]

Questions about Pythagorean triples, positive integer solutions to $a^2 + b^2 = c^2$.

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### Is there a formula that genrates only and all primitive Pythagorean triples?

The most common formula for generating Pythagorean triples is Euclid's, shown here as $$A=m^2-k^2\qquad B=2mk\qquad C=m^2+k^2$$ It generates all primitves but also generates trivals, doubles, square ...
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### New Pythagorean triples from old: a geometric proof?

There is a well-known construction that shows that in a right triangle with sides $a\le b<c$, we have $a^2+b^2=c^2$. Inscribe a square of side $c$ into a square of side $a+b$ so that each vertex of ...
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### Can we find other triangles with different integer sides which have equal integer area and perimeter using this method?

A few days ago, I was reading this Wikipedia page and this part caught my eyes: As W. A. Whitworth and D. Biddle proved in 1904, there are exactly three solutions, beyond the right triangles already ...
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### Results about generalised Pythagorean triples [duplicate]

I reduced a problem I am working on to the question of whether the equation $$x^2+y^2=6z^2$$ has a non-trivial integer solution. Thus I am looking for any results about the solution space of equations ...
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### Radii of touching spheres centered on Pythagorean triple vertices.

There are similar questions and answers out there but other answers I have seen seem unnecessarily complicated. I came up with what I think is a simple approach some time ago but I'm now doubting my ...
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### pythagorean triple factorization

Let $z>y>x$ primitive pythagorean triple. Show that $z$ has a prime factorization to primes of the form $4k+1$ I tried using the fact that there exist $s,t$ such that $s$ is not congruent to $t$ ...
1 vote
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### Constant Hiding Behind Particular Pythagorean Triples [duplicate]

The question pertains to an apparent constant appearing behind Pythagorean Triples whose "legs" have a difference of 1. To start, here are the first 5 triples that satisfy the above ...
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### Fairly simple problem: three integers, sums, and Pythagorean triples

So, I randomly thought about this simple problem, and very sadly couldn't make much progress in answering this. Does there exist three integers $a,b,c\neq0$ where $a^2+b^2$, $a^2+c^2$, and $b^2+c^2$ ...
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### Rotation matrix and Pythagorean theorem relation

From Wikipedia, the trace of the product of $R_z, R_y$, and $R_x$ is equal to $1 + 2\cos(\theta)$. Solving for $\theta$, we get $\theta$ = $\arccos\left(\frac{\text{trace} - 1}{2}\right)$. What is ...
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### If $x^2+px-q=0$ and $x^2+px+q=0$ have integer roots for real $p$ and $q$, then $p^2=a^2+b^2$ for some integers $a$ and $b$

Two equations $x^2+px-q=0$ and $x^2+px+q=0$ have integer roots, where $p$ and $q$ are real numbers. Prove that for some integers $a$ and $b$, $p^2 = a^2 + b^2$. In other words, have $\{a,b,p\}$ be a ...
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### What is the area of the shaded region if $AB=120$?

What is the area of the blue region if $AB=120$. I can only think in $\triangle OAB$ (O b being the low left point and the triangle is right) I can use Pythagoras which involves $120$, radius $r$ of ...
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### Can I infer the hypotenuse given only $a > b$?

Today, I came up with a problem. The problem is this: Let $a$, $b$ and $c$ be the sides of a right-angled triangle, such that $a > b$. The nature of $c$ is unknown. Can I infer the hypotenuse given ...
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### Gap between two Pythagorean triples

Let $s$ and $t$ be two positive integers. I am interested in the following Diophantine system : $$\mathscr{S}(s,t) : \left\{ \begin{array}{l} x^2+y^2=a^2\\ (x-s)^2+(y-t)^2=b^2 \end{array} \right.$$ ... 82 views

I'm struggling with a particular 3d problem - https://i.stack.imgur.com/SmOus.jpg (question 4) To workout the length of $EM$, forming a right angled triangle from face $EAB$ I get: $$EM =\sqrt{4^2 - ... 0 votes 1 answer 94 views ### Pythagorean "like" quadruples, help with general solutions. a long time ago I posted a question to find a general solution to a modified Pythagorean equation, mainly a^2+b^2=2c^2 that question was eventually answered. But now I need more help. I now have 3 ... 2 votes 3 answers 191 views ### Why can't a Pythagorean triple triangle have an odd area? To my understanding that a primitive triple x and y can be written as x = q^2 - p^2 while y=2pq for relatively prime opposite parity q > p then the area can be calculated as: pq(q^2 - p^... 3 votes 1 answer 180 views ### Parametrizations of  x^4+y^4+z^4=9t^2  integer solutions I would like to derive all the parametrizations for the nontrivial solutions of this Diophantine equation:  x^4+y^4+z^4=9t^2  I already know that with the Fauquembergue's parametrization I can find ... 2 votes 4 answers 180 views ### Show that there is no right triangle whose legs are rational numbers and whose hypotenuse is \sqrt{2022}. Show that there is no right triangle whose legs are rational numbers and whose hypotenuse is \sqrt{2022}. My tries: I used Pythagoras' Theorem to get:$$\sqrt{2022}^2=a^2+b^2 \implies a^2+b^2 = ... 246 views

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### Is a program that finds all pythagorean triples Involving one number useful?

I created a program that quickly finds all the pythagorean triples for one number (up to about 14 digits currently). Is this useful or is it fairly obvious to do? I'm a lay person in math but love ...
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### Question related to Bezout's identity and Pythagorean triples

Playing with Bezout's identity and Pythagorean triples, it seems that $a^{2}+b^{2}=c^{2}$ for coprime positive integers $a<b<c$ if and only if: $$bx+cy=a^{2}$$ $$b^{2}x+c^{2}y=1$$ Or which is ...
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### Berggren's Pythagorean ascent - reference request

Looking for an online resource and/or disambiguation for the following reference on Pythagorean triples: B. Berggren, Pytagoreiska trianglar, Tidskrift f¨or Element¨ar Matematik, Fysik och Kemi 17 (...
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### What is the Equation for every possible solution to the Pythagorean Theorem where a and b are x and y coordinates on a graph.

I currently am working on a minor programming project, and I have a program that can plot points in the problem above. However, this requires a lot of computing power that the average school ...
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### Why $(2m)^2 + (m^2 - 1)^2 = (m^2 + 1)^2$ results in pythagorean triples?

As you increase the value of n, you will generate all pythagorean triples whose first square is even. Is there any visual proof of the following explicit formula and where does it come from or how to ...
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### Integer solution of $a^2+b^2=c^2+d^2$ that relates $a,b,c$, and $d$ explicitly.

I have found that the integer solution of $a^2+b^2=c^2+d^2$ is $(a,b,c,d)=(pr+qs,ps-qr,pr-qs,ps+qr)$ for integer $p,r,q,s$. I wonder if there is an explicit relation between $a,b,c,$ and $d$? Or could ...
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### Is there a Pythagorean triple where the sum of the squares of all three members of the triple is itself a perfect square?

I am trying to create a problem for my students where they are essentially finding the length of a 3D vector. I wanted the problem to only use whole numbers/perfect squares. In the problem they are ...
I was working recently on Twin primes and circles. Thanks to Greg Martin, a nice generalization was conjectured in the comments section Consider the quarter-circle with center $0$ and radius $n$ or $... 2 votes 1 answer 76 views ### Euler Brick - Saunderson parametric solution Saunderson found a parametric solution giving Euler bricks: Let$(x,y,z)$be a pythagoran triple, then we get an Euler brick$(a,b,c)$with$a=x(4y^2-z^2)$,$b=y(4x^2-z^2)$,$c=4xyz$. It is claimed, ... 0 votes 1 answer 91 views ### How to prove a provided puzzle solution Below is a square that is$3 \times 3$match sticks. You are given 4 match sticks. Assume all match sticks to be of 1 unit dimension. The puzzle is to use 4 sticks along with the$3\times 3\$ square ... 