Questions tagged [pythagorean-triples]

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4
votes
3answers
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
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0answers
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
1answer
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
1answer
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
2answers
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
2answers
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
1answer
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
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0answers
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
1answer
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
2answers
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
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2answers
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
1answer
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
0answers
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
2answers
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
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2answers
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 - ...