Questions tagged [pythagorean-triples]
Questions about Pythagorean triples, positive integer solutions to $a^2 + b^2 = c^2$.
674
questions
4
votes
1
answer
85
views
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 ...
- 6,074
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 = ...
- 195
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 ...
- 6,074
0
votes
0
answers
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 ...
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=...
- 467
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 ...
- 3,645
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 ...
- 243
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 ...
- 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 ...
- 6,074
0
votes
2
answers
83
views
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 $t$ ...
- 211
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 ...
- 183
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 $b^2+c^2$ ...
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 ...
- 33
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 ...
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 ...
- 43
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 ...
- 33
1
vote
1
answer
65
views
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{ {...
- 17.8k
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? ...
- 25
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 ...
- 992
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 ...
- 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 (...
- 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 ...
- 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\,,...
- 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 :
$$\mathscr{S}(s,t) :
\left\{
\begin{array}{l}
x^2+y^2=a^2\\
(x-s)^2+(y-t)^2=b^2
\end{array}
\right.$$
...
user14479
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: $$EM =\sqrt{4^2 - ...
- 33
0
votes
1
answer
94
views
Pythagorean "like" quadruples, help with general solutions.
a long time ago I posted a question to find a general solution to a modified Pythagorean equation, mainly $a^2+b^2=2c^2$ that question was eventually answered. But now I need more help.
I now have 3 ...
- 179
2
votes
3
answers
191
views
Why can't a Pythagorean triple triangle have an odd area?
To my understanding that a primitive triple $x$ and $y$ can be written as $x = q^2 - p^2$ while $y=2pq$ for relatively prime opposite parity $q > p$ then the area can be calculated as: $pq(q^2 - p^...
- 356
3
votes
1
answer
180
views
Parametrizations of $ x^4+y^4+z^4=9t^2 $ integer solutions
I would like to derive all the parametrizations for the nontrivial solutions of this Diophantine equation:
$ x^4+y^4+z^4=9t^2 $
I already know that with the Fauquembergue's parametrization I can find ...
- 571
2
votes
4
answers
180
views
Show that there is no right triangle whose legs are rational numbers and whose hypotenuse is $\sqrt{2022}$.
Show that there is no right triangle whose legs are rational numbers and whose hypotenuse is $\sqrt{2022}$.
My tries:
I used Pythagoras' Theorem to get: $$\sqrt{2022}^2=a^2+b^2 \implies a^2+b^2 = ...
user1104319
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 + ...
user1135489
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 ...
- 6,074
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 ...
- 81
-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 ...
- 29
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,...
- 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^...
- 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 ...
- 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 ...
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 ...
- 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 $\...
- 2,417
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 ...
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 ...
- 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 (...
- 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 ...
- 11
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 ...
- 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 ...
- 93
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 ...
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 $...
- 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, ...
- 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 ...
- 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 ...
user1067672