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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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The Jacobi-Madden equation $a^4+b^4+c^4+d^4 = (a+b+c+d)^4$ and disguised Pythagorean triples

I. The Jacobi-Madden equation, $$a^4+b^4+c^4+d^4 = (a+b+c+d)^4$$ is equivalent to a disguised Pythagorean triple, $$(a^2+ab+b^2)^2+(c^2+cd+d^2)^2 = \big((a+b)^2+(a+b)(c+d)+(c+d)^2\big)^2$$ II. A ...
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1answer
179 views

Generating Pythagorean Triples from Others via Dissections

Roger Alperin's paper Modular Tree of Pythagoras shows it is possible to generate Pythagorean triples from others. If $a,b,c$ are the sides of a right triangle $a^2 + b^2 = c^2$ then we can derive ...
3
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1answer
74 views

Is it true that $n = 2t^2-2$ for even $t$ is a congruent number?

This post asks for $m$ such that the simultaneous Pythagorean triples, $$a^2+m^2b^2 = c^2\\b^2+c^2 = d^2\tag1$$ have solutions. Will Jagy found an infinite family given by, $$m = 2t^2-2 = 0, 6, 16, ...
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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
146 views

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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Prove (or provide a counterexample): no pair of primitive Pythagorean triples (a,b,c) and (2a,k,c) exists.

A primitive Pythagorean triple is an ordered set of coprime integers (a,b,c) such that $a^2+b^2=c^2$. Show that the system of Diophantine equations $$a^2+b^2=c^2$$ $$4a^2+k^2=c^2$$ have no solutions.
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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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$3$ primitive pythagorean triples from 6 integers.

Do there exist $3$ different primitive Pythagorean triples $(a,d,w), (a,b,z)$ and $(c,d,z)$? Explicitly, we want $6$ different integers $a,b,c,d,w,z$ such that... (1) $a^2 + d^2 = w^2$ (2) $a^2 + ...
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1answer
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Generating Pythagorean Triples From One Leg

The question is simple: Find the longest possible hypotenuse in a right triangle with integer sides where the shortest side has length T. What I am asking is if there are any means to approach this ...
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1answer
78 views

Pythagorean Triples $\mod{c}$

I have a quick question regarding modular arithmetic. If I have a Pythagorean Triple $(a, b, c)$, is it possible to consider this equation $\mod{c}$. That is to say, Is the implication $$a^2 +...
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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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1answer
56 views

How to find integer solutions an equation under a squareroot

I have a function $f(z) = \sqrt{(2z^2+1)^2+ 2^2 z(102z^2+151z+51)} $ I know it has some solutions where if $z$ and $f(z)$ is an integer, but are there infinitely many? or maybe is there some $z = f(...
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Need help with proof about Diophantine equations

The way I am planning to arrange this is by providing fragments of the proof, so I can understand what's going on before forging ahead, so if you are going to help me, keep in mind that I am going to ...
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1answer
29 views

Followup question about product of slopes on unit circle at rational points

This is a followup question to: Product of slopes of rational points on the unit circle (related to pythagorean triples) mathlove correctly showed that $D=1$ gives an infinity of solution pairs. But ...
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1answer
87 views

Pythagorean triplet .

You are given hypotenuse $h$ of triangle . Can you find out whether integral pythagorean triplet can be formed or not ? e.g given $h = 15$ . You can form triplet as $15,12,9$ because $15^2 = 12^2 + 9^...
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1answer
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How to find nonprimitive pythagorean triples given hypotenuse?

The integer $2015$ is the largest integer in $4$ different pythagorean triples, none of which is a primitive triple. Can someone explain how to find the $4$ triples?
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204 views

Geometric race line - 90 degree angular curve

I am in a process of solving the race curve by driving from outside to inside for $90˚$ curves. I'm trying to solve it with Pythagoras but somehow my theory seems to be flawed. Wolfphram alpha gives ...
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when $F_n^2+F_m^2$ is a square for fibonacci numbers

This is a curiosity question I'm trying to solve a Diophantine equation and I need some results about fibonnacci numbers, I encountered this problem: For which couple of integers $(n,m)$ the ...
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Conjectures on Twin Euler Bricks

After this answer euler bricks: way to calculate them? and this question Cuboid nearest to a cube I had some three hundred primitive Euler Brick solutions, complete with their Twin solutions (see ...
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Formulas for Pythagorean triples using Jacobsthal numbers and other well-known sequences?

Given Fibonacci numbers $F_n$, we have the known, $$(F_n F_{n+3})^2+(2F_{n+1}F_{n+2})^2 = (F_{2n+3})^2$$ as well as Lucas numbers $L_n$, $$(L_n L_{n+3})^2+(2L_{n+1}L_{n+2})^2 = (L_{2n+2}+L_{2n+4})^...
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Recognizing integer Pythagorean triples geometrically and whether two points constructed on a line are the same

This is suggested by the question Proof that the sum of the even side and the hypotenuse of a coprime (and positive) Pythagorean triple is a square number and the fact that I really like Euclid's ...
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Is this theorem Correct.

If it is, is it trivial? **Theorem** There does not exist a pythagorean triple $a^2 + b^2 = c^2$ $\{a,b,c \in \Bbb N\}$...
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Sum of the cubes of a Pythagorean triple equal a cube.

Apart from (3, 4, 5, 6) are there any more primitive solutions to $x^3+y^3+z^3=w^3$ where $x^2+y^2=z^2$ ? I’ve noted that if gcd(x ,y ,z) = k, then k divides w, so non-primitive Pythagorean triples ...
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441 views

Assumptions needed for proof of the Pythagorean Theorem from examples

There are lots of geometric dissection and reassembling proofs of the Pythagorean Theorem (PT). I want to know what are the necessary ideas to allow a proof using discrete instances. For example, we ...
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136 views

Pythagorean triple problem

I am doing research on perfect cuboids, and I'm looking for values $a,b,c$ such that the following is integer, and I'm not sure how to continue this. Any suggestions are appreciated! $PED$ is a very ...
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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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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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Can we find $n$ Pythagorean triples with a common leg for any $n$?

John Leech has a nice paper entitled, "Two Diophantine birds with one stone". The two birds in question are the two systems, $$t^2−3\big(a^2, b^2, (a + b)^2, (a−b)^2\big) = p^2, q^2, r^2, s^2$$ $$u^2 +...
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Solving $x^4+y^4+z^4 =t^2$ via elliptic curves

This post had an answer that involved the equation, $$x^4+y^4+z^4 =t^2$$ Holding $y,z$ constant, this is then a quartic polynomial in $x$ to be made a square. Using a birational transformation, it ...
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Rational distance between any pair of an infinite set on a specific circle

Is it possible to find an infinite set of point belonging to the circle of radius r such that $\sqrt{r} \notin \mathbb{Q}$ and where the distance between any pair is rational? For example, can we find ...
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143 views

Ternary Tree of Pythagorean triples

When following the approach to generate the ternary tree of Pythagorean triples with Fibonacci boxes, one has the root box \begin{bmatrix}1&1\\2&3\end{bmatrix} which corresponds to the ...
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How many “near-Fermat triples” are there?

Is the following statement true? Claim For every $n\in \mathbb N$, there is a constant $d$ such that there are infinitely many triples $a,b,c \in \mathbb N$ with $$ | a^n + b^n - c^n | \...
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$x^2+y^2=z^2$ in complex numbers

as a prelude to inquiring about solutions of Pythagoras' equation in Gaussian integers, it seemed sensible first to write out this equation for the complex case! i use the notation $z_i=x_i+iy_i$ and ...
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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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Proof By Descent FLT

So I've been studying the FLT and I am able to prove by descent that there are no solutions with $x,y,z$ natural numbers to $x^4+y^4=z^2$ I am trying to now prove by descent that there are no ...
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How can i solve this pythagorean triplet problem?

I came across this pythagorean triplet problem. According to which there is exactly one pythagorean triplet for which a+b+c = 1000 Pythagorean Triplet: a set ...
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Pairwise Pythagorean triples

Are there integers $a_1,a_2,a_3\in\Bbb N$ so that any two $a_i,a_j,i\not=j$ of them are the right side of a Phytagorean triple, i.e. $$a_i^2+a_j^2=b_{ij}^2,\qquad\text{for some }b_{ij}\in\Bbb N\quad?$...
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Method for Finding all Pythagorean Triplets including a number

I am aware of following technique to generate pythagorean triplets - $$ m^2 + n^2 , m^2 - n^2 , 2mn$$ However i have discovered a new technique which seems to be working as well - Lets say i want to ...
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Theorems of properties of rational right triangles proving Pythagorean triples

Let’s say we have a triple of integers $(a,b,c)$ which is assumed to be a Pythagorean triple, so that $$a^2+b^2=c^2.\tag{$\star$}$$ Without using ($\star$) directly or indirectly — but using other ...
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Distance geometry and pythagorean theory. Pairwise distances to absolute 2D coordinates

I don't have sufficient mathematical background. I am trying to get the absolute 2D coordinates from the pairwise comparison distances: What I have distances between points: p1-p2 = 0.3 p1-p3 = 0.5......
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43 views

Impossible form of a triangular number

Show that there are no positive integers $t,i,j$ with $j>i$ such that: $\displaystyle \frac{t(t+1)}2=\frac{2i(j-i)j(j+i)}3$ If possible provide an elementary proof. I believe the statement is ...
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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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70 views

Baffling Pythagoras Theorem Question

I came across this question. I asked my math teacher and the whole class yet no one came up with an answer. Could one of you try it? (The answers in the book say 3√2)
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Book describes some Pythagorean triples as uninteresting, favoring “primitive” Pythagorean triples. Why?

Background For reference: https://www.amazon.com/Friendly-Introduction-Number-Theory-4th/dp/0321816196 Chapter 2 starts with discussing Pythagorean triples (PT), which is an ordered triple $(a, b, c)...
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45 views

Getting a point from a point, length and angle

Using the diagram below, if we know the coordinates 'point A' and 'point B' as well as side lengths a,b,c. Is it possible to get the exact coordinates for points D & C that would also function as ...
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fermats last theorem for $p=83$

Prove the first case of fermat's last theorem for the prime $p=83$. If I let $\theta =2p+1$ then $\theta$ is $2\times83+1=167$ where $\theta$ is a prime. Computing their remainders on division by $83$...
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105 views

Construction of Pythagorean triples

If $(x,y,z)$ is a Pythagorean triple, then $\begin{pmatrix} 2 & 1 & 2 \\ 1 & 2 & 2 \\ 2 & 2 & 3 \end{pmatrix} \cdot \begin{pmatrix} x \\ y \\ z \end{pmatrix}$ is again a ...
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63 views

Complete formalization of solutions to $a^2+b^2=c^2+k$ for fixed $k>0$

Is there a known complete formalization of solutions to $a^2 + b^2 = c^2 + k$ for a fixed constant $k>0$ similar to the one for primitive Pythagorean triples (i.e. $(a,b,c) = (m^2-n^2,2mn,m^2+n^2)$ ...
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185 views

Pythagorean Triples using odd integers

I am trying to understand how the Proof by Contradiction can be applied to deriving the generalized rule for Pythagorean triples. I understand the general idea behind setting: $$\begin{align} a &=...
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28 views

Find two integers $a, b$ for given integer $c$, so that $c=a^2\pm b^2$

Given a positive integer $c$: Find two other positive integers $a$ and $b$, so that $c=a^2 + b^2$ and/or $c=a^2 - b^2$. I've already got a solution for any odd $c$: $c = (x+1)^2 - x^2 = 2x + 1$ so $...