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Questions tagged [pythagorean-triples]

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

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1answer
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Find all primitive Pythagorean triples such that all three sides are on an interval $[2000,3000]$

For primitive Pythagorean triples $(a,b,c)$, the following is valid: $$a=m^2-n^2,b=2mn,c=m^2+n^2$$ or $$b=m^2-n^2,a=2mn,c=m^2+n^2$$ $gcd(m,n)=1,m>n$ If numbers $a,b,c$ are relatively prime, then $...
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2answers
76 views

Meta-Pythagorean Triple

How can I find all Pythagorean triples $(a,b,c)$ such that the hypotenuse $c$ is a leg in another Pythagorean triple? For example, $(3,4,5)$ is such a Pythagorean triple because the length of the ...
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2answers
176 views
+50

About three conjectures concerning the Pythagorean Theorem.

I have two main conjectures relating to the Pythagorean Theorem that I desperately want to find out if they are true or not. Could somebody please help me? If $ \ a^2 + b^2 = c^2 \ $ for ...
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1answer
31 views

How to sensibly use Euclid's formula for Pythagorean triples. [on hold]

I've tried playing around with Euclid's formula ($A=m^2-n^2$, $B=2mn$, $C=m^2+n^2$) but I can't see any pattern in the triples it generates or how to predict what numbers will work other than being ...
6
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2answers
100 views

How to determine pythagorean triples that have a slope closest to 1

I'm not a mathematician, and I'm not sure how to phrase this question properly, so please bear with me as I stumble through the question. Considering Pythagorean's Theorem a²+b²=c² I'm looking for ...
1
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3answers
40 views

Sides of a Right-angled Triangle

$$(2n + 1)^2 + (2n^2 + 2n)^2 = (2n^2 +2n +1)^2$$ It can be used to generate infinitely many sides of right-angled triangles with integer lengths by putting values of $n = 1, 2, 3, ... $ I wanted to ...
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2answers
76 views

Triple Pythagorean with $a^2+b^2=c^4$

It is well known that there exist integer solutions to the equation $a^2+b^2=c^2$. For example, an explicit formula for integer values of $a$ , $b$ , and $c$ is \begin{align}a&=2mn \\ b&=m^2-...
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4answers
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Pythagorean Triples : Is every positive integer $\gt$ $2$ part of at least one Pythagorean triple?

I was doing some basic number theory problems from Rosen and came across this problem: Show that every positive integer $\gt$ $2$ is part of at least one ...
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1answer
33 views

Does an x exist so that there are n amount of pythagorean triples [closed]

Given $n\in\mathbb N$, does there exist an $x\in\mathbb N$, s.t. for $i\in\mathbb N,\;i\leq n$, $\exists y_i,z_i\in\mathbb N$ such that each $y_i$ is distinct and $$x^2 + y_i^2 = z_i^2$$
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1answer
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When does $\mathbb{Z}[\zeta_m]$ contain divisors of $2$ (besides units)?

Or equivalently, in which $\mathbb{Z}[\zeta_m]$ is $2$ reducible? And how does one construct any such divisors? $\bullet\ \textbf{My attempt}$ The smallest example seems to be $m=4$ with $2=(1+i)(1-...
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3answers
170 views

Is there a way of finding out the remaining two numbers of pythagorean triple if one of the side is given

I am solving one question related to right triangle and I have to find out the remaining two numbers of the pythagorean triple if one of the number is given. I know there can be many triples possible ,...
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3answers
390 views

Pythagorean Triple where $a=b$?

I cannot find even a single webpage mentioning this topic. I'm a programmer and I'm looking for a 45-45-90 triangle where all of the sides are whole numbers. In the video I am watching, they say to ...
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2answers
112 views

How to say that functions generate set members?

In a paper I am writing, I have seven abstract statements and, corresponding to the first one, I have a theorem statement that says I will prove that there is a Pythagorean triplet for every pair of ...
131
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1answer
4k views

Pythagorean triples that “survive” Euler's totient function

Suppose you have three positive integers $a, b, c$ that form a Pythagorean triple: \begin{equation} a^2 + b^2 = c^2. \tag{1}\label{1} \end{equation} Additionally, suppose that when you apply Euler's ...
2
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1answer
88 views

Matrix valued Pythagorean Triples

Consider any nxn matrices A, B and C such that A^2 + B^2 = C^2 Then the matrix triple (A,B,C) is called a Matrix valued Pythagorean Triple. I have observed that any nxn matrix M and N such that MN=NM,...
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1answer
126 views

What is the area of the quadrilateral $ADEC$ in $ABC$ right triangle in the following diagram?

In the right angled triangle $ABC$, $\angle A = 90^\circ$, $AB=8$, $AC=6$, $BC = 10$. $D$ is a point on $AB$ in such way that if a perpendicular $DE$ is drawn on $BC$ from $D$ then $BE = 4$. What ...
0
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1answer
19 views

Calculating length of triangle sides in trapezium

My younger brother has this mathematical problem to solve, and he came to me for help. At first I thought I could solve it by simply applying the Pythagorean theorem, but there seems to be more to it. ...
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3answers
35 views

Can we find two non-congruent right triangles with whole-number lengths and congruent hypotenuses?

I know some ways to find some Pythagorean triples. And I understand that if $a^2 + b^2 = c^2$ then $(a-b)^2 + (a+b)^2 = 2c^2$. I feel like that suggests a way forward, but I cannot find that way. ...
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2answers
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Length of an edge

My sister asked me to help her with her homework for mathematics, however frustratingly I was not able to figure out how to solve it. The assignment is as follows where it was requested to calculate ...
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1answer
43 views

Number of Pythagorean Triples

I am trying to solve an exercise from the book "Theory of Numbers" by B.M.Stewart. The exercise is the following one: Let $T=2^ap_1^{a_1}p_2^{a_2} \dots p_n^{a_n}$, where $a \ge0, n\ge0, 2<p_1&...
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4answers
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Derivation of Pythagorean Triple General Solution Starting Point:

I was reading on proof wiki about the derivation of the general solution to the pythagorean triple diophantine equation: $$ x^2 + y^2 = z^2, $$ where $x,y,z > 0$ are integers. I came across the ...
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5answers
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Can every perfect square exist as the sum or difference of two perfect squares?

I believe this is trivial and I'm over-complicating it. But can every squared integer be expressed as the sum of two squared integers OR the difference of two squared integers? And is there a proof ...
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2answers
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Did Babylonians know the Pythagorean theorem before his time?

On old tablets the Babylonians were able to work out the digits to the square root of two from the hypotenuse of a $45^\circ-45^\circ-90^\circ$ triangle. How could they have figured this out without ...
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3answers
77 views

Odd Pythagorian triplets [closed]

How many Pythagorean triplets $\{a,b,c\}$ exist, where $a,b,c$ are all odd? As far as I know there are no such triplets. $\{3, 4, 5\}; \{5,12,13\} ; \{7,24,25\}$ and its multiples are examples. Is ...
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1answer
92 views

Find all Pythagorean triangles whose area is twice a perfect square.

Find all Pythagorean triangles whose area is twice a perfect square. Let $x$ and $y$ be sides of a Pythagorean triangle. Using different approaches give no result! For example $xy=4d^2$ and $x^2+y^2=...
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3answers
64 views

Integer Solutions of the Equation $u^3 = r^2-s^2$

The question says the following: Find all primitive Pythagorean Triangles $x^2+y^2 = z^2$ such that $x$ is a perfect cube. The general solution for each variable are the following: $$x=r^2-s^2$$ $$...
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2answers
55 views

We can't exactly draw a line of length square root of 2 but can be constructed using Pythagoras theorem in an isosceles right angle triangle?

We can't exactly draw a line of length square root of 2 but in an isosceles right angle triangle of sides 1 unit each, the length of hypotenuse will be the square root of 2. Now does it mean we can ...
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1answer
40 views

Euler Brick calculation

How would you go about calculating Euler bricks from a list of primitive Pythagorean triples. I've tried to find an answer to this online but can't find anything which gives me list of Euler bricks ...
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2answers
94 views

Find all Pythagorean triples $x^2+y^2=z^2$ where $x=21$

Consider the following theorem: If $(x,y,z)$ are the lengths of a Primitive Pythagorean triangle, then $$x = r^2-s^2$$ $$y = 2rs$$ $$z = r^2+z^2$$ where $\gcd(r,s) = 1$ and $r,s$ are of opposite ...
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3answers
100 views

Pythagorean like Diophantine Equation

I am trying to solve this problem. http://www.javaist.com/rosecode/problem-527-1-2-3-type-Pythagorean-triangles-askyear-2018 Here we have to find all positive integral solution of $a^2+2b^2=3c^2$ ...
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2answers
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Prove that if $a, b, c \in \mathbb{Z^+}$ and $a^2+b^2=c^2$ then ${1\over2}(c-a)(c-b)$ is a perfect square.

Prove that if $a, b, c \in \mathbb{Z^+}$ and $a^2+b^2=c^2$ then ${1\over2}(c-a)(c-b)$ is a perfect square. I have tried to solve this question and did pretty well until I reached the end, so I was ...
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3answers
50 views

Pythagorean triple

Show that neither $1$ not $2$ can appear in any Pythagorean triple, but that every integer $k\geq3$ can appear. Prove that for each integer $k$ there are only finitely many Pythagorean triple ...
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3answers
441 views

For any primitive pythagorean triple $(a,b,c)$ either $a$ or $b$ must be a multiple of $3$ [duplicate]

I'm reading "Friendly Introduction to Number Theory". Now I'm working on Primitive Pythagorean Triples Exercises 2.1 (a) on P18. We showed that in any primitive Pythagorean triple $(a, b, c)$, ...
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1answer
52 views

Another proof for an infinite number of Pythagorean triples

I’m not sure if this has been mentioned before (and I truly apologize if someone thought about it already) , but I tried to adopt a geometrical approach for the proof for an infinite number of ...
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2answers
44 views

Basic questions about pythagorean triples and “n-lets”

I've had some difficulties finding answer to the two following questions: 1) Given one of natural numbers $a,b$ where $b$ is even and $a^2+b^2=c^2$ is there only one such a pythagorean triple? 2)How ...
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1answer
33 views

Integer solutions to AGM iteration

Any integer solution to $a^2+b^2=c^2$ also provides an integer solution $x=c$, $y=a$, $z=c+b$, $w=c-b$ to $$agm(x,y)=agm(z,w)$$ where $agm$ denotes Gauss' arithmetic-geometric mean. Are there other ...
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1answer
66 views

Prove for every $m, n$, $(m^2 - n^2, 2mn, m^2 + n^2)$ can make pythagorean triple

I tried researching more about this because it seems to be a common topic, but I don't know how to approach this problem. Do I have to somehow arrange those 3 terms into $a^2 + b^2 = c^2$?
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2answers
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A system of quadratic Diophantine equations with four variables

Is the following system has any positive integer solution $(x,y,u,v)$? $$\begin{cases} x^2+y^2=u^2\\ x^2-y^2=v^2 \end{cases}$$ I can prove that any pair of these integers can be relatively prime, but ...
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2answers
52 views

Generating Pythagorean triples

I'm asked to generate Pythagorean triples from the polynomial identity: $$(X^2-1)^2 + (2X)^2=(X^2+1)^2$$ By substituting rational numbers $\frac p q$ for $X$. However, Pythagorean triples are just as ...
2
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1answer
231 views

perfect right-triangles that are not super-perfect

I'm seeking some insight on the answer to this problem from Project Euler. Consider the right angled triangle with sides $a=7, b=24$, and $c=25$. The area of this triangle is $84$, which is ...
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1answer
45 views

Proof for General Properties of Pythagorean Triples

Having read the Wikipedia article "Pythagorean Triple", I came across the "Elementary properties of primitive Pythagorean triples" section which described many conditions for primitive triples, namely:...
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1answer
55 views

Do All Primitive Triples Belong to the Plato, Pythagoras and Fermat Families?

After running into this terminology on the "Formulas for generating Pythagorean triples" Wikipedia page, I was curious whether all triples fit into these categories. The article states: Plato: c - ...
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1answer
54 views

Can anyone help me with this Pythagoras Question?

The question is below Any help is appreciated sorry if I did something wrong this is just my first time using this. I tried to do it and I got either $\sqrt{31}$ or $\sqrt{32}$
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1answer
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Euler's totient function applied to higher power triples

I've been working my way through the mathematics presented in this question: Pythagorean triples that "survive" Euler's totient function concerning Pythagorean triples $a^2+b^2=c^2$ for ...
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2answers
103 views

Why there are exactly 100 distinct (not necessarily primitive) Pythagorean triples $(a,b,c)$ with $c<100$?

Why there are exactly 100 distinct (not necessarily primitive) Pythagorean triples $(a,b,c)$ with $c<100$? Using the fact that all primitive Pythagorean triples can be generated by the following: ...
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2answers
269 views

Is heron's formula inaccurate? [closed]

Here's one example of inaccuracy :- Suppose a triangle $XYZ$ with sides $a=13$, $b=15$ and $c= 14$. We have to find a perpendicular to side $c$ passing from point $X$. Image Link : https://i.stack....
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0answers
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Finding the area of an equilateral triangle using the Pythagorean theorem

From an equilateral triangle $T$ where each side have a length of $L$. What is the area of $T$? According to the Wikipedia page of equilateral triangles, the area is $$A=\dfrac{\sqrt{3}}{4}L^2$$ I ...
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5answers
3k views

Pythagorean theorem expressed without roots in an old Tamilian (Indian) statement

There's an old Tamil statement that predicts the hypotenuse of a right angle triangle to a reasonable level of accuracy considering it doesn't involve roots. This is how it goes: “Odum Neelam ...
10
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2answers
162 views

Solve $ \binom{a}{2} + \binom{b}{2} = \binom{c}{2} $ with $a,b,c \in \mathbb{Z}$

I am trying to solve the Diophantine equation: $$ \binom{a}{2} + \binom{b}{2} = \binom{c}{2} $$ Here's what it looks like if you expand, it's variant of the Pythagorean triples: $$ a \times (a-1) + ...
5
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4answers
714 views

How to descend within the “Tree of primitive Pythagorean triples”?

It is well-known that the set of all primitive Pythagorean triples has the structure of an infinite ternary rooted tree. What is the exact algorithm (i.e., formula, or possibly set of three formulas) ...