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 ...
poetasis's user avatar
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3 votes
2 answers
97 views

Are there infinitely many Pythagorean triples with two primes?

Using the classic parametrization of Pythagorean triples, for a primitive triple to contain two primes, we need $$ a = |m^2 - n^2|, c = m^2 + n^2 $$ to be prime. Taking $m<n$ WLOG, this means $a = ...
Dave Neary's user avatar
1 vote
0 answers
48 views

Vertex circle radii of Pythagorean triples

@Blue was kind with his comments on a previous question here. I'd now like to share some new relationships I found using algegra and my favorite formula for generating Pythagorean triples. In this ...
poetasis's user avatar
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0 votes
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36 views

Show that every primitive Pythagorean triple is of the form $(2 \lambda \mu, \lambda ^ 2 - \mu ^ 2, \lambda ^ 2 + \mu ^ 2)$

I am working through the book "Algebraic Geometry: A Problem Solving Approach" by Garrity et al., and I want somebody to check my work. I feel like I don't have a good grasp on projective ...
Hideki Miyukama's user avatar
0 votes
2 answers
61 views

If $a,b,c$ are nonzero natural numbers that verify the relation $a^2+b^2=c^2$, show that $a^n+b^n<c^n$. [duplicate]

PROBLEM If $a,b,c$ are nonzero natural numbers that verify the relation $a^2+b^2=c^2$, show that $a^n+b^n<c^n$, where $n$ is a strictly natural number greater than $2$. WHAT I THOUGHT OF $a^2+b^2=...
Ionela Buciu's user avatar
1 vote
0 answers
63 views

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 ...
Alexander Burstein's user avatar
0 votes
1 answer
62 views

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 ...
Etack Sxchange's user avatar
0 votes
0 answers
37 views

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 ...
Schief's user avatar
  • 321
2 votes
0 answers
46 views

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 ...
poetasis's user avatar
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0 votes
2 answers
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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 $t$ ...
Tamir Vered's user avatar
1 vote
0 answers
30 views

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 ...
Connor James's user avatar
0 votes
1 answer
49 views

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 $b^2+c^2$ ...
TheMather - or rather AMather's user avatar
2 votes
1 answer
70 views

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 ...
Promisek3u's user avatar
5 votes
1 answer
233 views

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 ...
geniuspig2986's user avatar
4 votes
1 answer
128 views

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 ...
Yesmar 's user avatar
2 votes
1 answer
59 views

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 ...
tf2-doc's user avatar
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1 vote
1 answer
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What is the maximum number of points $p_i\in\mathbb{Z}^2$ such that no three points are co-linear and $d(p_i,p_j)\in\mathbb{Z}\forall i,j\in[k]?$

What is $k,$ the maximum number of points $p_i = (x_i, y_i)\in\mathbb{Z}^2,\ i\in \{1,\ldots,k\},\ $ such that no three points $p_{i_1},\ p_{i_2},\ p_{i_3},\ $ are co-linear, and $d(p_i,p_j):= \sqrt{ {...
Adam Rubinson's user avatar
1 vote
1 answer
101 views

Can 3 pythagorean triangles on the same circle have areas where two of the areas sum to the third?

Let’s say we have 3 pythagorean triangles (integer sides and hypotenuse) all with the same hypotenuse. Is it possible for the areas of two of those triangles to sum to the area of the third triangle? ...
ServingSpy's user avatar
2 votes
1 answer
116 views

Finding rational points on a circle such that $X^2+Y^2=r^2=k \in \mathbb{Z}$

I am interested in finding rational points on a circle with radius $r$, such that $r^2=k$ is an arbitrary integer. I tried reducing the problem to the unit circle, and maybe use pythagorean triples as ...
MC From Scratch's user avatar
2 votes
1 answer
122 views

Finding points an integer distance from all vertices of a triangle

I am working through some old questions from the Swiss Mathematics Games. Specifically, I am trying to solve problem 18 from the 35th Quarter Finals (from 2021). The question and solutions can be ...
Uzai's user avatar
  • 497
3 votes
2 answers
113 views

How to prove the existence of a Pythagorean Triple without finding solutions?

I am looking to prove that there is a Pythagorean Triple (x, y, 173) without finding solutions. I know that a solution does exist ((52, 165, 173)), however I would like to prove this more generally (...
Azorbz's user avatar
  • 185
0 votes
1 answer
82 views

Distribution distance ; Riemann Geometry; Information Geometry

As the upper image shows, $M_1$ is a distribution whose parameter is θ, while $M_2$ corresponds to $θ'=θ-δg$. $δ$ means learning rate and $g$ means the gradient. $M_2$ is optimised from $M_1$ by ...
1 1's user avatar
  • 5
0 votes
2 answers
142 views

How does opposite $\sin60\unicode{176}$ have $0.866\ldots$ if hypotenuse is $1\,?$ [closed]

I've approached in a few different ways to prove it, but couldn't get any good results. When : $d = 2\,,$ $\text{radius}=\dfrac d2\;\rightarrow\;r=1\,.$ Since hypotenuse is the same length with $\,r\,,...
l3lue's user avatar
  • 111
10 votes
5 answers
497 views

Gap between two Pythagorean triples

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

How can I work out the angle between a face and base of a triangular prism

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 - ...
Morgan's user avatar
  • 33
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 ...
spydragon's user avatar
  • 179
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^...
gus f's user avatar
  • 356
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 ...
user967210's user avatar
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 = ...
user avatar
4 votes
1 answer
246 views

Why can’t Pythagorean triples of this form exist, $\{a, b, c\}$, $\{2a, d, c\}$?

Why can’t there exist two Pythagorean triples with the same hypotenuse, but one has a leg that is twice as long as a leg of the other? Or why is there no integer solution to $a^2 + b^2 = c^2$ $4a^2 + ...
user avatar
0 votes
2 answers
555 views

Distance from a right angle to a hypotenuse.

Given a right triangle $\,(A,B,C)\,$ with legs $x$ and $y,\,$ there is a point $h$ on the hypotenuse dividing the latter into $h_1$ (from the origin to $h$), and $h_2$ (from $h$ to the end of the ...
poetasis's user avatar
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0 votes
2 answers
83 views

Finding an example of a Pythagorean triple, such that $\gcd(x,y,z)=1$ but $\gcd(x,z)>1$, $\gcd(x,y)>1$, and $\gcd(y,z)>1$

Finding an example of a Pythagorean triple $x^2+y^2=z^2$, with the $\gcd(x,y,z)=1$ but $\gcd(x,z)>1$, $\gcd(x,y)>1$, and $\gcd(y,z)>1$. In order for this to be the case then I need to have an ...
cheesewiz's user avatar
-1 votes
1 answer
132 views

Prove that N is a perfect square

The integer $N$ is positive. There are exactly $2005$ ordered pairs $(x, y)$ of positive integers satisfying: $\frac{1}{x}$ + $\frac{1}{y}$ = $\frac{1}{N}$ Prove that $N$ is a perfect square. Please ...
iimonchii's user avatar
0 votes
0 answers
63 views

Prove that the integers solutions to $x^4-2y^2=1$ are only $(1,0)$ and $(-1,0)$ [duplicate]

I want to prove that the integers solutions to $x^4-2y^2=1$ are only $(1,0)$ and $(-1,0)$ There is a hint: The positive primitive solutions of $x^2 + y^2 = z^2$ with $y$ is even are $x=r^2-s^2, ~y=2rs,...
Gang men's user avatar
  • 399
0 votes
0 answers
40 views

Find the representation of solutions of $x^2+5y^2=z^2$ when $\gcd(x,y)=1$ [duplicate]

I want to show that for $x^2+5y^2=z^2$ and $\gcd(x,y)=1$, there are 2 cases: $x$ is odd and $y$ is even: then $x=\pm(r^2+5s^2),~y=2rs,~z=r^2+5s^2$ $x$ is even and $y$ is odd: then $x=\pm(2r^2+2rs-2s^...
Gang men's user avatar
  • 399
2 votes
4 answers
158 views

Can 2021 be hypotenuse of a pythagorean triangle?

I'm looking for Pythagorean triangles with a hypotenuse of length 2021. In other words, let $z=2021$ then, $z^2=x^2+y^2$ for some integers $x$, $y$. Now, when I put a hypotenuse value of 2021 to ...
Meow's user avatar
  • 152
1 vote
1 answer
197 views

Pythagorean quintuple $x^2+y^2+z^2+w^2=u^2$ has infinite solutions. Is it difficult to find the general formula of its integer solution?

Pythagorean quintuple equation $x^2+y^2+z^2+w^2=u^2$ has the infinite set of integer solutions, but it seems that no general solution formula has been published. There are methods to give more ...
X. D. Dongfang's user avatar
5 votes
4 answers
171 views

Pythagorean triples, proof clarification

Question Prove for every odd integer $a \geq 3$ that there exists an even integer $b$ such that $(a, b, b + 1)$ is a Pythagorean triple. Proof Let $a \geq 3$ be an odd integer. Then $a = 2n + 1$ for ...
John Doe's user avatar
  • 502
4 votes
3 answers
199 views

Approximating $\sqrt{2}$ by Pythagorean triples? [duplicate]

I am trying to find sequence of Pythagorean triples $(x_{n}, y_{n}, z_{n})\in\mathbb{Z}^{3}$ such that $x_{n}/y_{n}\rightarrow 1$. This way, both $z_{n}/x_{n}$ and $z_{n}/y_{n}$ would converge to $\...
Maximal Ideal's user avatar
0 votes
1 answer
54 views

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 ...
pattern_mancer's user avatar
0 votes
1 answer
44 views

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 ...
Juan Moreno's user avatar
  • 1,032
0 votes
0 answers
39 views

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 (...
vvg's user avatar
  • 3,133
-1 votes
1 answer
86 views

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 ...
Michael Lutz's user avatar
1 vote
5 answers
120 views

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 ...
YGranja's user avatar
  • 161
3 votes
2 answers
116 views

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 ...
ZeroToZero's user avatar
0 votes
1 answer
90 views

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 ...
David Obeid's user avatar
2 votes
1 answer
98 views

Pythagorean triples, primes and circles.

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 $...
vengy's user avatar
  • 1,649
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, ...
xbut123's user avatar
  • 53
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 ...
jonadv's user avatar
  • 125
-2 votes
2 answers
81 views

Generating Sequence of Right angled triangle

A "nearly isoceles" right-angled triangle with integer side lengths is defined as one in which the two sides adjacent to the right angle differ in length by just 1 unit. A triangle with side ...
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