Questions tagged [pythagorean-triples]

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

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Pythagorean triples-number theory [closed]

How can I prove that the hypotenuse $h$ of a primitive Pythagorean triangle (that is a right triangle which sides are given by a primitive Pythagorean triple) can be written as $h=12k+1$ or $h=12k+5$...
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Does it follow from the Pythagorean theorem that “there is no right-angled triangle with sides 5, 10 and 11”?

Assume, for the sake of this question, that we define the Pythagoeran theorem as Theorem 1: In a right-angled triangle, the sum of the squares of the two shortest sides is equal to the square of the ...
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Pythagorian triples condition

Pythagorian triple is every triple of natural numbers $(x, y, z)$ such that $x, y, z$ are sides of right triangle, where $z$ is hypotenuse. Now, Pythagorian theorem says $x^2 + y^2 = z^2.$ (1) If we ...
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Is there another formula which generates Pythagoras' triples such that the largest $2$ of the triple differ by $3$?

I was thinking about Pythagoras' triples recently, and I wondered if I could find a formula that generated Pythagorean triples such that the largest of $2$ numbers of the triple differ by $1$, and I ...
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Reference request: there are only two integer solutions to $2^{2a} + 3^{2b} = 5^c$. [duplicate]

I believe there are only two non-negative integer solutions to $$2^{2a} + 3^{2b} = 5^c.$$ The solutions I have are $a=1,b=0,c=1$ and $a=2,b=1,c=2$. I'm not certain this is correct. I'd like to know if ...
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prove if $a$ is odd then there exists a Pythagorean triplet with it [duplicate]

I want to show that if $1 < a \in \mathbb{N} $ is odd, then there exists $b,c\in \mathbb{N}$ s.t (a,b,c) is a primitive Pythagorean triplet. I believe it would be enough to show there exists $s<...
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Pythagorean Triples using points on the circle

It's given a circle with center $(0,0)$ and radius $1.$ There are four points on the circle (see the graph). How to form Pythagorean triples using these points? I assume that I have to use Inscribed ...
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1answer
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$\frac{1}{a} = \frac{1}{b} + \frac{1}{c} - \frac{1}{abc}$ and $a^2 + b^2 = c^2$

I have found this in an Romanian magazine. We have to solve for natural numbers: $$\frac{1}{a} = \frac{1}{b} + \frac{1}{c} - \frac{1}{abc}$$ $$a ^ 2 + b ^ 2 = c ^ 2$$ After some elementary ...
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Finding the sides of a Triangle

So i've tried this question for a while now, but can't seem to get an answer. I tried to equate z but don't know how to proceed. Can someone help? In△XYZ, the measure of∠XZY is 90. Also, YZ=x cm, XZ=y ...
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Finding integer solutions to the system $xy=6(x+y+z)$, $x^2+y^2=z^2$

How do you go about solving a system of equations like below for integer solutions? $$xy=6(x+y+z)$$ $$x^2+y^2=z^2$$ Would you first try and list out a number of Pythagorean triples, then try and see ...
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Let P be a point inside an equilateral $△ABC, PA=5, PB=12, PC=13$. What is the ratio in which P divides the 3 sections of the triangle?

We can find the edge length of the triangle, see this thread. My approach for the triplets are to find a $90° + 60°$ i.e. see $△XPB$ , $∠XPB = 90$ & $∠APX = 60°$. Image Followed by using the ...
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Proving right isosceles triangle

The right triangle XYZ with sides of lengths x and y, and hypotenuse of length z satisfies that z=√(2xy) , and the triangle XYZ is an isosceles. (This is my answer and I want to know your comments ...
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Is a Pythagorean triple uniquely determined by its smallest element? [closed]

A triple $(a,b,c)$ is a Pythagorean triple if $a,b$ and $c$ are strictly positive integers satisfying $a^2 + b^2 = c^2$. Do there exist $a,b,c,b',c'$ for which $(a,b,c)$ and $(a,b',c')$ are both ...
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Trigonometry Grade 10 question [closed]

If $$(m^2-n^2)\sin \theta+2mn \cos \theta =m^2+n^2$$ then find $\tan\theta$. Please use trigonometric identities of Grade 10
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Given $\textrm{gcd}(p,q)=1$, in what cases do positive integers $k$ and $a$ exist for which $k^2 + (2q)^2 = (aq+p)^2$?

Given $\textrm{gcd}(p,q)=1$, in what cases do positive integers $k$ and $a$ exist for which $k^2 + (2q)^2 = (aq+p)^2$? I have tried to approach this problem using Euclid's parameterization of ...
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Are my calculations of Pythagorean Triplets including 0 correct?

So I was playing around with Pythagorean Triplets and I decided to see what happens when certain sides are equal to 0. For a² + b² = c², If a = 0, Since (2m)² + (m² - 1)² = (m² + 1)², m = 0 Therefore, ...
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Calculating the intersection points of two circles, of which the centers are located on the x axis.

I'm trying to solve the problem mentioned in the title, and the answer detailed here provides exactly what I was looking for: https://stackoverflow.com/a/3349134/4777480...except for the fact that the ...
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A Question on Integers and Fractions

Given that $a,b,c,d,e,f$ are all distinct positive integers, prove that there are no solutions for the following knowing their fractions are non-integers and completely simplified: $$\left(\frac{a}{c}\...
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A Class of Diophantine Equations in Three Variables

This question is inspired by the problem https://projecteuler.net/problem=748 Consider the Diophantine equation $$\frac{1}{x^2}+\frac{1}{y^2}=\frac{k}{z^2}$$ $k$ is a squarefree number. $A_k(n)$ is ...
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Pythagorean-Hodograph curve's control points

In the Farouki's book "Pythagorean-hodograph curves: algebra and geometry inseparable" it is said that control points of the cubic Bezier curves with Pythagorean-hodograph qualities are set ...
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Simplest Proof of Euclid's Formula for Primitive Triples?

A right triangle with side lengths $a,b,c$ where $c$ is the hypotenuse and using integers $m, n $ where $ m > n$, we can find Euclid's Formula. First, given a right triangle with an hypotenuse $\...
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Generating a New Set of Pythagorean Triples?

For those it may concern, if you have seen a similar post before, only the start remains similar. Using a right triangle with side lengths $(a,b,c)$ where $a , b < c$, I was thinking about how the ...
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Is there a tree of Pythagorean quadruples?

It is well-known that there is a trinary tree that contains every primitive Pythagorean triple exactly once. It even has a Wikipedia page. Is there a similar tree that contains each primitive ...
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Is this Pythagorean triple formula original?

I am hesitant because I have asked this question before in a different form here. In 2009, I knew nothing about Pythagorean triples, not even Euclid's formula. In my ignorance, I did a brute force ...
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1answer
118 views

Generating Pythagorean Triples with a New Method?

Using a right triangle with side lengths $(a,b,c)$ where $a < b < c$, I was thinking about how the area of a Pythagorean triple can be found using the Pythagorean triple right before it and I ...
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1answer
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How many right angled triangles with co-prime integer sides and base of length $28cm$ are there?

How many right angled triangles with co-prime integer sides and base of length $28cm$ are there? Please help me. My working: I tried assuming that once that base side is $2m = 28$ as $ 2m,m^2-1,m^2+1 ...
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When does the distance traveled in vertical and horizontal lines become the hypotenuse? [duplicate]

I am looking for a formal proof for the following problem: You travel from point A to point B on a right triangle only along its legs. For a 3 4 5 right triangle you would travel a distance of seven. ...
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1answer
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Generating Euler Bricks

In a problem set assigned to my class, we were asked to show that if $(u,v,w)$ is a Pythagorean triple, then a cuboid with side lengths $u|4v^2-w^2|$, $v|4u^2-w^2|$, and $4uvw$ generated an Euler ...
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1answer
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Does the incenter divide the triangle into a ratio of 2:1?

I'm trying to get the radius of the circle. First I use the Pythagorean theorem $13^2=5^2+x^2$ and $x=12$. Then I know the height of the triangle. To get the radius i use the Pythagorean theorem $r^2=...
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For what integer values of $k$ does $a^{2}+b^{2}=kc^{2}$ have positive integer solutions?

The obvious case of $k=1$ clearly has solutions, for example, $(3,4,5)$ and $(5,12,13)$, among others. Trivially, it has infinitely many solutions by multiplying through by any number. It has ...
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Finding Pythagorean triples, with a conjecture.

A right triangle with integer side lengths ($a , b , c$ where $a < b < c$) can be found (to a certain degree) with the inner radius ($r$), the formulas for the area of a triangle and Pythagoras ...
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4answers
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Finding Pythagorean triple where the squares of sides are weighted

Suppose $a, b, c \in \mathbb{Z}_{>0}$. The following is a variation of a Pythagorean triple, but with weighted squares: $$a^2 + 3 b^2 = 4 c^2$$ Can I find something like Euclid's formula that can ...
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Solving parametric in rationals

I want to find all values of a such that the equation $$\sqrt{a - x} = a - x^2$$ has at least one real root and none of its roots are irrational. I made some decent ...
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Estimate percent of Pythagorean triples where $A>B$

Over a decade ago, I developed a formula that generates the subset $S$ of Pythagorean triples where $GCD(A,B,C)$ is an odd square. (Note: This includes all primitive where $GCD(A,B,C)=1.)\quad$ This ...
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Alternative to Heron's Formula , through Proof by Pythagorean Theorem.

(This works for all triangles) The area ($A$) of a right triangle, can be given by the inner radius ($r$) multiplied by the semi perimeter ($s$), where $r = \frac12(a + b- c)$ and $s = \frac12(a + b +...
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How is this 20square root3 here?

The question that we are given a polygon on left on diagram with each side 20m. Then in the right side I made a triangle of ABC. I drew head and tail of the vector on my own. Angle I got between ...
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1answer
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Find exactly $3$ matching primitive Pythagorean triples for a given hypotenuse

I'm trying to find three Pythagorean triples $\quad A^2+B^2=C^2 \quad\text{where}$ $$A_1^2+B_1^2=A_2^2+B_2^2=A_3^2+B_3^2=C^2\quad \text{and} \\ A_1\ne A_2,\ne A_3\quad\land \quad B_1\ne B_2,\ne B_3$$ ...
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Is there a perfect square number n, whose Euler Totient value is also a perfect square?

Start with a perfect square, denoted as a positive integer n. The root of this square is k, another positive integer. Thus n = k^2 Let t = totient(n) Is there a way to prove there is no such number? I ...
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If $a,b,c \in \mathbb{Z}$ and $a^2+b^2=c^2$ show that $abc \equiv 0 \pmod{60}$. [duplicate]

If $a,b,c \in \mathbb{Z}$ and $a^2+b^2=c^2$ show that $abc \equiv 0 \pmod{60}$. I once read, on a number theory textbook - forget the title, in one of the problems list that all Pythagorean triplets ...
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1answer
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Shortest distance for cuboid by taking a certain path in 3D [duplicate]

I am stuck on part (b) of the question. Please see image above. How would I work out the shortest distance with reducing the amount of cheese the ant has to crawl through?
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Height of a cube edge from the floor

A cube $ABCD.EFGH$ has side length $2a$ cm. Point $A$ is lifted $a$ cm from the floor, point $C$ is still on the floor, point $B$ and point $D$ are on the same height from the floor. What is the ...
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Solving Dudeney's “A Question of Cubes”: sum of consecutive cubes is a square

The following is a puzzle of Dudeney: Professor Rackbrane pointed out one morning that the cubes of successive numbers, starting from $1,$ would sum to a square number... He stated that if you are ...
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Is every triangle with 3 rational angles (in degrees) similar to one with 3 integer-length sides?

The same way right triangles have pythagorean triples, does every non-right class of similar triangles also have some integer triples, if we say that all the angles have to be rational numbers in ...
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The smallest number that can be discribed as a sum of three squares in two different ways with equal multiple.

What is the smallest number $k$ that is a sum of three squares in two different ways(there does not exist a pair of numbers from different triple which are the same) $$a_0^2+b_0^2+c_0^2=a_1^2+b_1^2+...
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1answer
122 views

Higher Order Diophantine Equation

I am trying to obtain the general solution of the following Diophantine equation. $$x^4-8x^2y^2+8y^4=z^2$$ My approach was to rewrite the equation. \begin{align} \begin{split} & x^4+z^2=2(2y^2-x^2)...
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2answers
41 views

How does this hypotenuse/derivative/polynomial approximation work?

I am trying to understand an article regarding the forces of a vibrating string in terms of both longitudinal and transverse waves. Here is the pertinent excerpt: Equation 1 is obvious. It is just ...
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Consecutive Integers Pythagorean Triplets

I have a question for which I was not able to find an answer online. I was wondering how many Pythagorean Triplets we have found till now which consists of three consecutive integers like $(3,4,5)$.
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42 views

Prove that if (a,b,c) is a Pythagorean triple, then 3 divides either $a$ or $b$ without remainder classes [duplicate]

This number theory book I have proved that if $(a, b, c)$ are a primitive Pythagorean triple (i.e., $a, b, c$ have no common factors and $a^2 +b^2 = c^2$), then either $a$ or $b$ is odd. The book did ...
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1answer
79 views

How many zeros does $x^2 + y^2 \pmod d$ have on $[0, d-1]^2$?

I have been doing some work on Pythagoras's Theorem with my Year 8 Maths class (7th Grade in US speak). I had them investigating what values were unobtainable for the square on the hypotenuse of right-...

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