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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
45 views

How to sensibly use Euclid's formula for Pythagorean triples.

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
101 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 ...
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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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2answers
77 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-...
3
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1answer
43 views

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-...
0
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1answer
147 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 ...
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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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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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2answers
113 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 ...
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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
36 views

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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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
65 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
57 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 ...
0
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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 ...
2
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2answers
95 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 $a^2+2b^2=3c^2$ ...
7
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2answers
55 views

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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1answer
54 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
99 views

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 ...
3
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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 ...
1
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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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3answers
448 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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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
237 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
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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
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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
121 views

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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3answers
112 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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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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1answer
90 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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2answers
170 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) + ...
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2answers
90 views

Are there any 2 primitive pythagorean triples who share a common leg?

So is it possible for: $\gcd(a,b,c)=1$ $a^2+b^2=c^2$ and $\gcd(a,d,e)=1$ $a^2+d^2=e^2$ ?
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1answer
162 views

Help prove $p^4 + 4q^4$ is never a perfect square?

Given that $p$ and $q$ have no common factors, how can I prove that $p^4 + 4q^4 \not =z^2$, if $p,q$ and $z$ have to be positive integers? From the comments: I'm working on the congruent number ...
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2answers
38 views

How to obtain the bounds for a and b?

If $a,b,c$ are Pythagorean triplets. Provided $a+b+c = s$ and $a<b<c$. We can conclude that $a\le\frac{s-3}{3}$ and $b<\frac{s-a}{2}$. From the given conditions we can obtain: $a^{2} + b^{2}...
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1answer
138 views

Nontrivial integer solutions of $ a^3+b^3=c^3+d^3$

How can we obtain a set of nontrivial solutions of $$ a^3+b^3=c^3+d^3, $$ for $a,b,c,d\in \mathbb{Z}$ where $(a,b)\neq (c,d)$ and $(a,b)\neq (d,c)$. Say in the range that $|a|,|b|,|c|,|d| \in [...
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1answer
33 views

Explain this n-modulus relation in Pythagorean triplets {a,a+n,c}

I've found a seemingly consistent Pythagorean triplet relation for which I cannot find a proof or counterexample. Theorem: For all Pythagorean triplets {a, b, c} where $gcd(a,b)=1$, b-a is odd and 2 ...
2
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1answer
45 views

When can an odd integer $d$ be represented as $d=a^2-2b^2$ with coprime integers $a,b\ $?

I found out that in a primitive pythagorean triple $$a^2+b^2=c^2$$ the difference $d=|a-b|$ (which must be odd) can occur, if and only if we can write $$d=a^2-2b^2$$ with positive coprime integers $a,...
12
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4answers
431 views

Explain this convergence among Pythagorean triplets

Why do the ratios of successive values of integers $a$ and $c$, where $a^{2}+(a+1)^{2}=c^{2}$, appear to converge to $$\frac{a_{n+1}}{a_{n}},\frac{c_{n+1}}{c_{n}}\rightarrow3+2\sqrt{2}$$ I rigorously ...
2
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0answers
37 views

Totient values of consecutive positive integers forming a pythagorean triple

Suppose $a$ is a positive integer. When do the totient values of $a$ , $a+1$ and $a+2$ form a pythagorean triple ? In other words : For which positive integers $a$ does the equation $$\phi(a)^2+\...
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1answer
84 views

Do any (non-trivial) 2-chains of Pythagorean triplets exist?

Define an "integer 3-relationship" as a function $f(a,b,c)$ of three integer variables, together with the condition that this function must equal zero. Two examples would be the "Pythagorean triplet ...
10
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2answers
130 views

If $a+b+c$ divides the product $abc$, then is $(a,b,c)$ a Pythagorean Triple?

Firstly, I will define what Pythagorean Triples are for those who do not know. Definition: A Pythagorean Triple is a group of three integers $a$, $b$ and $c$ such that $a^2+b^2=c^2$, ...
0
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1answer
60 views

Can the Pythagorean Theorem be extended like this? [closed]

What is the significance, if any, of the fact that $3^3$ + $4^3$ + $5^3$ = $6^3$ ? How curious this is. This would be like the Pythagorean Theorem exploding into a higher dimension, on steroids.
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2answers
47 views

The effect of scalars on primitive Pythagorean triples

When some scalars are applied to primitive Pythagorean triples, the triple can still be expressed in terms of m and n with x=m^2-n^2 y=2mn z=m^2+n^2 but with others it changes it. So far I've ...
2
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0answers
42 views

Does a representation $d=a^2-2b^2$ with coprime integers always exist?

I studied which differences $a-b$ are possible in primitive pythagorean triples $(a/b/c)$. I noticed that the difference $d$ must be odd and contains only prime factors with quadratic residue $2$, ...