Skip to main content

Questions tagged [pythagorean-triples]

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

Filter by
Sorted by
Tagged with
2 votes
2 answers
71 views

Tiling potential of the primitive triplets

What is the maximum percentage of an infinite plane that could be tiled with primitive Pythagorean triangles, if every triangle may be used at most once? My intuitive guess is that it can’t reach 100%,...
Nalacram's user avatar
-1 votes
1 answer
49 views

Pythagorean triangles with non rectilinear [closed]

Does there exist a Pythagorean triple whose corner is non-rectilinear with axes, yet all three vertices are on integer coordinates? The question was closed due to "lack of context". The ...
Dwayne Towell's user avatar
0 votes
1 answer
80 views

Imaginary Units in the Pythagorean Theorem

Are there any exceptions to the Pythagorean Theorem? I ask this after seeing the triangle with sides i, $1$, and $0$. Sure, this triangle checks out with the Pythagorean Theorem ($i^2+1^2=0^2$), but ...
Grey's user avatar
  • 739
2 votes
2 answers
111 views

Distribution of primes in primitive Pythagorean triples

My Observation: I've observed a pattern where for every pair of twin primes ($p$, $p+2$), there appears to be at least one primitive Pythagorean triple ($a$, $b$, $c$) such that one of the twin primes ...
Nicholas Joseph's user avatar
10 votes
3 answers
384 views

Integer Right Triangle with Repdigit Area

This question was inspired by a tweet by Cliff Pickover. I'm seeking a right triangle with integer sides, and whose area is a repdigit number greater than $666666$. I know that Pythagorean Triples can ...
DreiCleaner's user avatar
  • 1,569
0 votes
2 answers
73 views

Finding the Pythagorean triples for a given small side

I wish to find all Pythagorean triples containing a given number as one of it's small sides. Take a right angled triangle $a,b,c$ small side $a$ and hypotenuse $c$. Let $N=c-b$ Substituting into ...
Mark Lester's user avatar
5 votes
1 answer
304 views

Find prime numbers satisfying an equation

Find all triplets $(m, n, p)$, where $p$ is a prime number and $m, n ∈ \Bbb N$, such that $p=\frac{m}{4}\sqrt{{2n-m \over 2n+m}}$ My procedure is as follows: $p=\frac{m}{4}{\sqrt{(2n)^2-m^2}\over 2n+m}...
Pranav P J's user avatar
4 votes
2 answers
100 views

Example of Complex Pythagorean Triples

I am looking for example of a Pythagorean Triple with Gaussian Integers. I followed the links and looked at followings : Relation to Gaussian integers in https://en.m.wikipedia.org/wiki/...
jimjim's user avatar
  • 9,748
-1 votes
1 answer
128 views

Seeking opinions about a formula for OEIS

I proposed a formula $a(n,k) = |A-B|=|j^2-2*k^2|, j=(2*n-1),\quad n,k \in N, GCD(j,k)=1,$ for the OEIS series Numbers whose prime factors are all congruent to +1 or -1 modulo 8. $( 1, 7, 17, 23, 31, ...
poetasis's user avatar
  • 6,416
1 vote
1 answer
105 views

Why does the term being squared in the Pythagorean triplet make no difference to the whole triplet?

I was recently preparing for the SOF IMO Level 2 exam. While practising for the exam, I encountered this question in my textbook about the chapter Square and Square Roots which also mentions the ...
Sambhav Khandelwal's user avatar
1 vote
0 answers
74 views

Pythagoras theorem twice [duplicate]

Is there any pair of positive integers,$a,b,c,d$,that $a^2+b^2=c^2$ and $b^2+c^2=d^2$ holds true? Thanks!
A Math guy's user avatar
3 votes
1 answer
125 views

Four rational Pythagorean triples in a square

Suppose we have a unit square $ABCD$. If we have a primitive Pythagorean triple $(a, b, c)$, let $CD = 1$ and hence $ED = a/b$. It also follows that $EC$ is rational of course. In fact, using the ...
Toby M's user avatar
  • 113
-1 votes
2 answers
103 views

How to find the side lengths of a triangle in the incircle of a Pythagorean triple

Given a Pythagorean triple with an incircle, how do I find the sides of a triangle connecting the triple's tangents to that incircle. In the diagram below, I know how to find side $\,i\,$ but not the ...
poetasis's user avatar
  • 6,416
3 votes
0 answers
81 views

Matrix preserving almost Pythagorean triples

The following matrix is known to generate Pythagorean triples given an initial solution $(x,y,z) $ by recursive matrix multiplication. $$ H = \begin{bmatrix} 1 & 2 & 2 \\ 2 & 1 & 2 \\ ...
numbergobbler's user avatar
-1 votes
2 answers
326 views

Proving that there is a sequence of $2023$ distinct integers such that the sum of the squares of any two consecutive terms is a perfect square itself [closed]

How to prove the following problem? Prove that, there is a sequence of $2023$ distinct integers such that the sum of the squares of any two consecutive terms is a perfect square itself. I tried with ...
Rajan saha Raju's user avatar
-2 votes
1 answer
199 views

Can every congruent number be written as $\dfrac{nm(n-m)(m+n)}{\delta}$ [closed]

I want to show every congruent number $\alpha $ be written as $$\alpha = \dfrac{nm(n-m)(m+n)}{\delta}$$ for some $m,n$ where $\gcd(m,n)=1$, $m\not \equiv n\mod 2 $, $m>n$ and $\delta$ a divisor of $...
MrPie 's user avatar
  • 235
1 vote
1 answer
97 views

Does each proper congruent number correspond to some Pythagorean triple? [duplicate]

Define a proper congruent number as an integer which represents the area of some right triangle $\triangle ABC$ whose legs are both strictly in $\mathbb{Q}/\mathbb{Z}$. For every proper congruent ...
MrPie 's user avatar
  • 235
0 votes
2 answers
60 views

Are there any semi congruent numbers?

Define a semi congruent number $\,n\,$ as a congruent number which represents the area of a triangle $\,\bigtriangleup ABC\,$ which has one leg in $\,\mathbb{Q}\setminus \mathbb{Z}\,$ and the other in ...
MrPie 's user avatar
  • 235
2 votes
1 answer
105 views

Connection of Perpendicular in right triangle

I am sorry first, my mother language is not English so I have a problem to explain ‘what I want to know’. But I will try my best. The Pythagorean rule said that “C^2=ab”. And I understood if a=b, ...
HYOUNG GYU PARK's user avatar
1 vote
1 answer
68 views

Is my reasoning right about an extreneous root?

In another post, using this formula for generating Pythagorean triples \begin{align*} &A=(2n-1+k)^2-k^2&&=(2n-1)^2+2(2n-1)k\\ &B=2(2n-1+k)k &&=\phantom{(2n-1)^2+{}} 2(2n-1)...
poetasis's user avatar
  • 6,416
-4 votes
1 answer
195 views

Is there any odd $n\in \mathbb{N}$ satisfying $n=e^2-f^2=g^2-h^2=k^2-l^2$ and $e^2f^2=g^2h^2+k^2l^2$ where $e, f, g, h, k, l~(>1)\in \mathbb{N}$?

Is there any odd natural number $n$ satisfying $n=e^2-f^2=g^2-h^2=k^2-l^2$ and $e^2f^2=g^2h^2+k^2l^2$ where $e, f, g, h, k, l~(>1)$ are natural numbers? The problem is related to famous open ...
From God's Sanatan Country's user avatar
5 votes
3 answers
170 views

When is $x^2+9y^2 \pm y$ a perfect square

I have come across a problem in which it is to be determined for what values of $x$ the expression $x^2+9y^2 \pm y$ is a perfect square, i.e. $x^2+9y^2 \pm y=z^2$, for $x,y,z \in \mathbb N$ There are ...
Keith Backman's user avatar
0 votes
1 answer
143 views

Why Pythagorean triple leg differences have only certain values.

A variation of Euclid's formula seems to reveal why primitive Pythagorean triple leg differences appear to be restricted to the number $\,1,\,$ to selected primes raised to any integer power, or to ...
poetasis's user avatar
  • 6,416
5 votes
2 answers
247 views

Proving Exhaustion of Primitive Pythagorean Triples

I claim that the following algorithm will always give a primitive Pythagorean triple. Let $d$ be some even number and let $D$ be the set containing all its factors. Then for any two coprime factors $...
Leonidas Lanier's user avatar
6 votes
2 answers
359 views

To get all the (b,a,a+1) Pythagorean triplets [closed]

In $(b,a,a+1)$ I found that $a+a+1=b^2$, in which $a>b$ You guys probably knew this but I randomly thought this thing in $5,12,13$ and then found it in all with $b,a,a+1$ So anyways That means $2a+...
VIBHU's user avatar
  • 73
0 votes
1 answer
106 views

Is it possible to prove that if $x$ and $y$ are co-prime, then $(x-y)$ and $\sqrt {xy}$ are also co-prime?

I was trying to prove that pythagorean triplets exists in Natural Number domain. Here's simplified argument that I did: consider two natural numbers $x$ and $y$ such that $x > y \ge 1$. $(x + y)^...
Cinverse's user avatar
  • 181
1 vote
0 answers
42 views

Confused about a Wikipedia explanation on primitive Pythagorean triplets

Link to the Paragraph: "Proof of Euclid's formula" https://en.wikipedia.org/wiki/Pythagorean_triple#:~:text=Proof%20of%20Euclid%27s%20formula In this paragraph (please read from 4th sentence ...
Cinverse's user avatar
  • 181
4 votes
1 answer
96 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 ...
poetasis's user avatar
  • 6,416
3 votes
2 answers
173 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
53 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
  • 6,416
0 votes
0 answers
44 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
65 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
65 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
68 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 ...
user avatar
0 votes
0 answers
38 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
  • 319
2 votes
0 answers
50 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
  • 6,416
0 votes
2 answers
101 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 $t$ ...
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
53 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
129 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
87 views

Is the set of “Pythagorean” square roots dense in $\left[0, \infty\right)$? [closed]

I was thinking about proofs of the fact that square roots exist and ended up wondering if the set of numbers of the form $\sqrt{\dfrac{a^2}{b^2}+ \dfrac{c^2}{d^2}}$ with $a, b, c, d \in\Bbb N$ is a ...
skh's user avatar
  • 55
5 votes
1 answer
290 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
174 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
60 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
  • 33
1 vote
1 answer
67 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{ {...
Adam Rubinson's user avatar
1 vote
1 answer
121 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
255 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
145 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
168 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
99 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

1
2 3 4 5
15