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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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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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How is it that a third line segment doesn't always divide the first two?

How can it be shown by the pythagorean theorem that it's not always possible to find a third line segment that evenly divides into the first two? I'm using the unit square as an example. Does this ...
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1answer
82 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
61 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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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
34 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
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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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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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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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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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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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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
32 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
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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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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
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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 ...
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1answer
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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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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
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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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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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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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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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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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149 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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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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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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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
133 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
31 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 ...
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1answer
42 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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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 ...
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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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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 ...
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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 $a^2+b^2=c^2$, ...
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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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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 ...
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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 $2$, ...
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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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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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1answer
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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??
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In a unit circle, what allows the same trigonometric rules to be applied past the first quadrant?

I had previously asked this question and was asked to learn more about the unit circle, which I've done. I now have further questions: When the concept of trigonometric ratios for acute angles is ...
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2answers
63 views

No primitive pythagorean triple $(x,y,z)$ with $z\equiv -1 \pmod 4$

Show that there are no primitive pythagorean triple $(x,y,z)$ with $z\equiv -1 \pmod 4$. I once have proven that, for all integers $a,b$, we have that $a^2 + b^2$ is congruent to $0$, or $1$, or $2$ ...
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Are there any (more) consecutive numbers that are functional in the Pythagorean Theorem?

The numbers three, four, and five can be implanted in the Pythagorean process (3=a, 4=b, 5=c) to equal a correct right triangle. But my question is, are there any more directly consecutive numbers ...
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Given primitive Pythagorean triple $(a,b,c)$ with $a<b<c$, if $a$ is odd, then $a^2 = b+c$

I've noticed that, for any pythagorean triple $(a,b,c)$ arranged least to greatest, if they have no common divisor (i.e., if the triple is primitive) and $a$ is odd, then following holds: $$a^2=b+c$$ ...
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3answers
36 views

Formula for the the integers of all Pythagorean triples with $z = y + 2$

Find formulas for the integers of all Pythagorean types $x,y,z$ with $z = y + 2$ I know $z = m^2 + n^2$ and $y = 2mn$ so substituting I get... $m^2 + n^2 = 2mn + 2 \Rightarrow (m-n)^2 = 2$ which ...
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116 views

The Proof of Infinitude of Pythagorean Triples $(x,x+1,z)$

Proof that there exists infinity positive integers triple $x^2+y^2=z^2$ that $x,y$ are consecutive integers, then exhibit five of them. This is a question in my number theory textbook, the given hint ...
2
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2answers
55 views

Finding the area of a triangle in a trapezoid and the area of the trapezoid based on the given information.

An isosceles trapezoid $ABCD$ has bases $AD = 17$cm, $BC = 5$cm, and leg $AB = 10 $cm. A line is drawn through vertex $B$ so that it bisects diagonal $AC$ and intersect $AD$ at point $M$. 1) Find ...
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3answers
210 views

Non-pythagorean triples

Euclid's formula gives a recipe for generating all possible Pythagorean triples $a^2+b^2 = c^2$ exactly once; for set of positive integers $k$ and $m>n$ ($m$ relatively prime to $n$, exactly one of ...