Questions tagged [pythagorean-triples]
Questions about Pythagorean triples, positive integer solutions to $a^2 + b^2 = c^2$.
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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
85 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 ...
1
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3answers
35 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 ...
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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2answers
106 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
34 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 ...
0
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1answer
41 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
75 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
88 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=...
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:
$$x=r^2-s^2$$
$$...
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2answers
49 views
We can't exactly draw a line of length square root of 2 but can be constructed using Pythagoras theorem in an isosceles right angle triangle?
We can't exactly draw a line of length square root of 2 but in an isosceles right angle triangle of sides 1 unit each, the length of hypotenuse will be the square root of 2. Now does it mean we can ...
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1answer
40 views
Euler Brick calculation
How would you go about calculating Euler bricks from a list of primitive Pythagorean triples. I've tried to find an answer to this online but can't find anything which gives me list of Euler bricks ...
2
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2answers
91 views
Find all Pythagorean triples $x^2+y^2=z^2$ where $x=21$
Consider the following theorem:
If $(x,y,z)$ are the lengths of a Primitive Pythagorean triangle, then $$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
99 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
votes
2answers
54 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
49 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
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
94 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 ...
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2answers
43 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)$$ where $agm$ denotes Gauss' arithmetic-geometric mean. Are there other ...
0
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1answer
62 views
Prove for every $m, n$, $(m^2 - n^2, 2mn, m^2 + n^2)$ can make pythagorean triple
I tried researching more about this because it seems to be a common topic, but I don't know how to approach this problem. Do I have to somehow arrange those 3 terms into $a^2 + b^2 = c^2$?
2
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3answers
431 views
For any primitive pythagorean triple $(a,b,c)$ either $a$ or $b$ must be a multiple of $3$ [duplicate]
I'm reading "Friendly Introduction to Number Theory". Now I'm working on Primitive Pythagorean Triples Exercises 2.1 (a) on P18.
We showed that in any primitive Pythagorean triple $(a, b, c)$, ...
1
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2answers
49 views
Generating Pythagorean triples
I'm asked to generate Pythagorean triples from the polynomial identity:
$$(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
211 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
42 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
52 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
53 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
119 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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2answers
100 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
70 views
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$$
I ...
2
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1answer
77 views
Matrix valued Pythagorean Triples
Consider any nxn matrices A, B and C such that
A^2 + B^2 = C^2
Then the matrix triple (A,B,C) is called a Matrix valued Pythagorean Triple.
I have observed that any nxn matrix M and N such that MN=NM,...
10
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2answers
156 views
Solve $ \binom{a}{2} + \binom{b}{2} = \binom{c}{2} $ with $a,b,c \in \mathbb{Z}$
I am trying to solve the Diophantine equation:
$$ \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) + ...
2
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2answers
87 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
$\gcd(a,d,e)=1$
$a^2+d^2=e^2$
?
0
votes
1answer
154 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
137 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
votes
1answer
43 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,...
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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 ...
3
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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+\...
5
votes
1answer
80 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
127 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
57 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.
1
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2answers
46 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
votes
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$, ...
10
votes
3answers
305 views
Matrix version of Pythagoras theorem
Can I find a solution for $C_{n\times n}$, explicitly, for the given $A_{n\times n}$ and $B_{n\times n}$ such that
$AA^{T} + BB^{T} = CC^{T}$? Here $A^{T}$ denotes the transpose of $A$ and all the ...
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votes
2answers
227 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....
0
votes
1answer
23 views
Proving divisibility involving Pythagorean triplet.
Let $a$, $b$, $c$ be natural numbers such that
$$ a^2 +b^2=c^2 $$ and $$c-b=1.$$
Prove that $$a^b + b^a $$ is divisible by $c$.
Any hints??
