Questions tagged [pythagorean-triples]
For questions about Pythagorean triples which are triples of positive integers $(a, b, c)$ satisfying $a^2 + b^2 = c^2$.
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What is the connection between these three methods of generating this sequence?
I was recently looking at this problem: “There are a number of balls in a jar, some of them red, some of them white. The odds of picking two at random and both balls being red is 1/2. How many of the ...
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We have $\binom{62}{26}^2+\binom{62}{27}^2=\binom{62}{28}^2$. How many other Pythagorean triples are contained in a single row of Pascal's triangle?
At MSE I asked, "Does any row of Pascal's triangle contain a Pythagorean triple?" The answer is yes; the example $\binom{62}{26}^2+\binom{62}{27}^2=\binom{62}{28}^2$ was given. In that ...
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Can the Pythagorean Graph be finitely colored?
Define the Pythagorean Graph as having nodes $a,b\in \mathbb{N}_{\ge 3}$ and an edge $a\rightarrow b$ if and only if $a^2+b^2$ is a square. After much searching I found the example in the picture, ...
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Reference request on a pattern among nearly isosceles Pythagorean triples
Let us understand the term nearly isosceles Pythagorean triple to mean one whose legs differ by $1.$ A fortiori such a triple is primitive.
After someone asked me how to find such triples, it was easy ...
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Size of set of integers with all sums of two distinct elements giving squares
Are there arbitrarily large sets $\mathcal S=\{a_1,\ldots,a_n\}$ of strictly positive integers such that all sums $a_i+a_j$ of two distinct elements in $\mathcal S$ are squares?
Considering subsets in ...
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Pythagorean triples and quadratic residues modulo primes
QUESTION. Are my following conjectures true? How to prove them?
Conjecture 1. For each prime $p>100$, there are $a,b,c\in\{1,\ldots,p-1\}$ such that
$$\left(\frac ap\right)=\left(\frac bp\right)=\...
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Finding Pythagorean quadruples on a given plane?
In 2D one cannot construct Pythagorean triples $x^2+y^2=m^2$ ($x,y,m\in\mathbb{Z}$) that lie on every line through the origin (e.g., a Pythagorean triple with $x=y$ would require $\sqrt{2}$ to be ...
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Number of integers in relation to Pythagorean triples with some quadratic relations
Given an integer $m>0$ possibly composite we can find non-negative integers which are not equivalent to $0\bmod m$ with
$$ab+cd\equiv0\bmod m$$.
Is there any integer quadruples bounded in $[0,m-1]^...
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Small linear relations between primitive Pythagorean triples $\mathsf{II}$
WillJagy answered a linear relation question on Pythagorean Triples in Small linear relations between primitive Pythagorean triples $\mathsf I$.
Now let $a^2+b^2=c^2$ be a primitive Pythagorean ...
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Small linear relations between primitive Pythagorean triples $\mathsf I$
Say $a^2+b^2=c^2$ is a primitive Pythagorean triple. Then consider the Linear Diophantine Equation
$$ua^2+vb^2+xab+ybc+zca=0$$ where $(u,v,x, y, z)\in\mathbb Z^4$ are variables. If $(u,v,x, y, z)\neq(...
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Triangulating the plane using edges of unique rational lengths
Basic question: Can the Euclidean plane be divided into a vertex-to-vertex arrangement of non-overlapping triangles such that every edge has a unique rational length that lies between 1 and some ...
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Prove $\frac{\text{Area}_1}{c_1^2}+\frac{\text{Area}_2}{c_2^2}\neq \frac{\text{Area}_3}{c_3^2}$ for all primitive Pythagorean triples
A while ago I asked this question on MSE here. After placing a bounty it got quite a bit of attention but unfortunately it has yet to be resolved. After getting some advice from MO Meta I have decided ...
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Сomplement of the set of numbers of the form $ 4mn - m - n$?
Numbers of the form $4mn-m-n$ where $m,n\in\mathbb{Z}^+$ are
$$
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, ...
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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 ...
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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
...
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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 ...
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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 $j\geq5$; ...
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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 ...
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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&...
user76479
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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?
user76479
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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 - ...
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