# Questions tagged [pythagorean-triples]

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15
questions

**4**

votes

**3**answers

237 views

### Finding Pythagorean quadruples on a given plane?

In 2D one cannot construct Pythagorean triples $x^2+y^2=m^2$ ($x,y,m\in\mathbb{Z}$) that lie on every line through the origin (e.g., a Pythagorean triple with $x=y$ would require $\sqrt{2}$ to be ...

**1**

vote

**0**answers

74 views

### Number of integers in relation to Pythagorean triples with some quadratic relations

Given an integer $m>0$ possibly composite we can find non-negative integers which are not equivalent to $0\bmod m$ with
$$ab+cd\equiv0\bmod m$$.
Is there any integer quadruples bounded in $[0,m-1]^...

**4**

votes

**1**answer

174 views

### Small linear relations between primitive Pythagorean triples $\mathsf{II}$

WillJagy answered a linear relation question on Pythagorean Triples in Small linear relations between primitive Pythagorean triples $\mathsf I$.
Now let $a^2+b^2=c^2$ be a primitive Pythagorean ...

**1**

vote

**1**answer

149 views

### Small linear relations between primitive Pythagorean triples $\mathsf I$

Say $a^2+b^2=c^2$ is a primitive Pythagorean triple. Then consider the Linear Diophantine Equation
$$ua^2+vb^2+xab+ybc+zca=0$$ where $(u,v,x, y, z)\in\mathbb Z^4$ are variables. If $(u,v,x, y, z)\neq(...

**5**

votes

**2**answers

272 views

### Triangulating the plane using edges of unique rational lengths

Basic question: Can the Euclidean plane be divided into a vertex-to-vertex arrangement of non-overlapping triangles such that every edge has a unique rational length that lies between 1 and some ...

**16**

votes

**2**answers

844 views

### Prove $\frac{\text{Area}_1}{c_1^2}+\frac{\text{Area}_2}{c_2^2}\neq \frac{\text{Area}_3}{c_3^2}$ for all primitive Pythagorean triples

A while ago I asked this question on MSE here. After placing a bounty it got quite a bit of attention but unfortunately it has yet to be resolved. After getting some advice from MO Meta I have decided ...

**3**

votes

**1**answer

222 views

### Сomplement of the set of numbers of the form $ 4mn - m - n$?

Numbers of the form $4mn-m-n$ where $m,n\in\mathbb{Z}^+$ are
$$
A=\{2, 5, 8, 11, 12, 14, 17, 19, 20, 23, 26, 29, 30, 32, 33, 35, \ldots\}
$$
The set complement of the above set is
$$
B=\{1, 3, 4, 6, ...

**7**

votes

**0**answers

277 views

### Is 100 the only Leyland number that is a square?

Leyland numbers (named for Paul Leyland) are positive integers of the form $x^y + y^x$ , where x and y are naturals > 1, and also the number 3. The OEIS link is https://oeis.org/A076980
I thought ...

**8**

votes

**1**answer

331 views

### Integer points avoiding three on a line, four on a circle

A century ago, Dudeney asked to place $16$ pawns on a chessboard with no three
on a line:
As described by David Eppstein,1 the maximum number $g_3(n)$ points that
...

**5**

votes

**2**answers

400 views

### Can we surround a non-rectangular area with Lego fences?

My children have some Duplo fences, these you have to put down on two points, and at both ends they extend a little where you can connect several to surround some area.
So a fence is described by a ...

**8**

votes

**2**answers

1k views

### Is there any formula to find number of Pythagorean triplets between two integers 2 and j, j>2?

Given $j \geq 5$, is there a formula for the number of Pythagorean triplets $(a, b, c)$ satisfying the constraint that $a, b, c \leq j$?
There exists at least one Pythagorean triplet for $j\geq5$; ...

**13**

votes

**1**answer

565 views

### Realization of numbers as a sum of three squares via right-angled tetrahedra

De Gua's theorem
is a $3$-dimensional analog of the Pythagorean theorem:
The square of the area of the diagonal face of a right-angled tetrahedron
is the sum of the squares of the areas of the other ...

**2**

votes

**0**answers

101 views

### Primitive triples in a region [duplicate]

Are there at least $cn$ Pythagorean triples and at least $dn$ Primitive Pythagorean triples $(A,B,C)$ with $$2^{\frac n2}<A<2^{\frac n2+1}<2^n<B<C<2^{n+1}$$
with some fixed $0<d&...

**1**

vote

**2**answers

212 views

### Sharply Estimating Pythagorean Triples [closed]

Given $m,n\in\Bbb N$ with $m<n$, how many pythagorean triples $p^2+r^2=q^2$ satisfy $$m\leq p<r\leq n?$$
Is there a way to give a sharp estimate?

**17**

votes

**2**answers

2k views

### Assistance with understanding parent/child relationships in Pythagorean Triples

I want to start by apologising for what is probably a weak attempt at a question on a site like this, but I'm having trouble understand a concept that doesn't seem to be properly explained elsewhere - ...