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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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Pythagorean triples that “survive” Euler's totient function

Suppose you have three positive integers $a, b, c$ that form a Pythagorean triple: \begin{equation} a^2 + b^2 = c^2. \tag{1}\label{1} \end{equation} Additionally, suppose that when you apply Euler's ...
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6answers
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Prove that the sum of pythagorean triples is always even

Problem: Given $a^2 + b^2 = c^2$ show $a + b + c$ is always even My Attempt, Case by case analysis: Case 1: a is odd, b is odd. From the first equation, $odd^2 + odd^2 = c^2$ $odd + odd = c^2 \...
23
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5answers
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Can two perfect squares average to a third perfect square? [duplicate]

My question is does there exist a triple of integers, $a<b<c$ such that $b^2 = \frac{a^2+c^2}{2}$ I suspect that the answer to this is no but I have not been able to prove it yet. I realize ...
20
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5answers
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A very different property of primitive Pythagorean triplets: Can number be in more than two of them?

While playing with numbers, I thought about squares of numbers, and then the first thing that came to mind was Pythagorean triplets. I observed a very interesting fact that any $x\in\mathbb N$ can ...
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5answers
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Does any given integer only occur in one primitive Pythagorean triple?

I know that all integers are part of at least one primitive triple. But can this statement be refined to exactly one? From looking at some lists of triples it seems to be true, but I have no clue ...
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16
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2answers
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$x^2+y^2=z^n$: Find solutions without Pythagoras!

I was presented with the following problem: Prove that there exist solutions to $x^2+y^2=z^n$ for all $n$, with $x,y,z, n \in \mathbb{N}$ I showed that by taking any Pythagorean triple $x^2+y^2=z^...
13
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2answers
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Fermat's Last Theorem for Negative $n$

While studying Fermat's Last Theorem and Pythagorean triples, the following question occurred to me: For the equation $a^n+b^n=c^n$, where $n$ is a negative integer, a) does a solution exist, and b) ...
13
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1answer
137 views

Pythagorean triplets of the form $a^2+(a+1)^2=c^2$ and the space between them

I was searching for pythagorean triples where $b=a+1$, and I found using a python program I made the first 10 integer solutions: $0^2+1^2=1^2$ $3^2+4^2=5^2$ $20^2+21^2=29^2$ $119^2+120^2=169^2$ $696^...
12
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3answers
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Can $\sqrt{ab}$ and $\sqrt{a^2 + b^2}$ be both integers if $a$ and $b$ are natural numbers?

Does there exist an $a \in \mathbb{N}$ and $b \in \mathbb{N}$ such that $\sqrt{ab} \in \mathbb{Z}$ and $\sqrt{a^2 + b^2} \in \mathbb{Z}$?
12
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4answers
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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 ...
12
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2answers
498 views

Show divisibility by 7

I was stuck at this question: Suppose $a^2+b^2=c^2$ for $a,b,c \in \mathbb Z$, and neither $a$ nor $b$ is a multiple of 7. Show that $a^2-b^2$ is a multiple of 7 I tried to write $b^2$ as $c^2-a^2$...
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0answers
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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 ...
11
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2answers
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Why can no prime number appear as the length of a hypotenuse in more than one Pythagorean triangle?

Why is it that no prime number can appear as the length of a hypotenuse in more than one Pythagorean triangle? In other words, could any of you give me a algebraic proof for the following? Given ...
11
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2answers
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Is the hypotenuse of a triangle ever divisible by three (for primitive Pythagorean triples)?

Looking for a proof that for primitive Pythagorean triples, the hypotenuse is never divisible by three. Below are a list of all the primitive Pythagorean triples with a hypotenuses less than 300. ...
11
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3answers
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For which $n$ are there primitive Pythagorean triples with legs of lengths $a$ and $a+n$?

For which n can $a^{2}+(a+n)^{2}=c^{2}$ be solved, where $a,b,c,n$ are positive integers? I have found solutions for $n=1,7,17,23,31,41,47,79,89$ and for multiples of $7,17,23$... Are there ...
10
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6answers
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Are there finitely many Pythagorean triples whose smallest two numbers differ by 1?

Has it been shown whether there is a finite or infinite number of Pythagorean triples whose smallest two numbers differ by 1? In either case I’d appreciate a link to the proof. Edit: thank you all ...
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5answers
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Pythagorean theorem expressed without roots in an old Tamilian (Indian) statement

There's an old Tamil statement that predicts the hypotenuse of a right angle triangle to a reasonable level of accuracy considering it doesn't involve roots. This is how it goes: “Odum Neelam ...
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2answers
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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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3answers
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If $\gcd(x,y)=1$, and $x^2 + y^2$ is a perfect sixth power, then $xy$ is a multiple of $11$

This is a problem that I don't know how to solve: Let $x, y, z$ integer numbers such that $x$ and $y$ are relatively primes and $x^2+y^2=z^6$ . Show that $x\cdot y$ is a multiple of $11$.
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2answers
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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) + ...
10
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3answers
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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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3answers
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Is there a Pythagorean triple whose angles are 90, 45, and 45 degrees?

Is there a Pythagorean triple (a.k.a. an integer triangle) whose angles are 90, 45 and 45 degrees? I am trying to connect LEGO roads at angles other than the standard 90 degrees.
9
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2answers
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Can a right triangle have odd-length legs and even-length hypotenuse?

Is it possible to have an even integer hypotenuse and odd integer legs (perpendicular and base) in a right triangle? If yes, please give an example. If no then please prove that.
9
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4answers
561 views

How can you find a Pythagorean triple with $a^2+b^2=c^2$ and $a/b$ close to $5/7$?

How can you find a Pythagorean triple with $a^2+b^2=c^2$ and $a/b$ close to $5/7$? I've been reading the Plimpton 322 news, and this fits in the gap in the Babylonian table between 0.6996 ($a=1679,b=...
9
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4answers
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Sum of $2$ equal squares also a square

Is there an integer solution to $a^2 + a^2 = b^2$? Because there's this universift that has this logo of the pytagorean theorem where the two squares are equal, but I don't think it's possible.
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3answers
363 views

Three pythagorean triples

Are there any solutions for $a, b, c$ such that: $$a, b, c \in \Bbb N_1$$ $$\sqrt{a^2+(b+c)^2} \in \Bbb N_1$$ $$\sqrt{b^2+(a+c)^2} \in \Bbb N_1$$ $$\sqrt{c^2+(a+b)^2} \in \Bbb N_1$$
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2answers
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Is it well known that for the Pythagorean spiral, $\sum_1^\infty\frac1{a_n+1} = \frac12$?

Let $\{a_n:n\in\Bbb Z^+\}$ be the sequence defined by $a_1=3$ and for $n\in\Bbb Z$ $$a_{n+1} = \frac12(a_n^2+1) $$ Note that all the $a_n$ are odd integers. This is called the "Pythagorean Spiral," ...
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3answers
585 views

Finding $n$ satisfying that there is no set $(a,b,c,d)$ such that $a^2+b^2=c^2$ and $a^2+nb^2=d^2$

Let us consider $n\ge 3\in\mathbb N$ which satisfy the following condition. Condition : There exist no set of four non-zero integers $(a,b,c,d)$ such that $$a^2+b^2=c^2\ \ \text{and}\ \ a^2+nb^2=d^2.$...
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2answers
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Is there a way to find a pythagorean triple so that when you place a given digit before it, it still is a pythagorean triple?

For example, in base 10: $$5^2 + 12^2 = 13^2$$ And when I put a one before each number, the equality still holds: $$15^2 + 112^2 = 113^2$$ So my question is, in a given base, is there a way to get ...
9
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2answers
487 views

Quadruple of Pythagorean triples with same area

Can one find explicitly $a_i,b_i,c_i\in\Bbb N,i=1,2,3,4$ so that $$ a_i<b_i, \qquad \text{ and } \qquad a_i^2+b_i^2=c^2_i \qquad\text{for } i=1,2,3,4$$ and $$a_1b_1=a_2b_2=a_3b_3=a_4b_4, \...
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4answers
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Derivation of Pythagorean Triple General Solution Starting Point:

I was reading on proof wiki about the derivation of the general solution to the pythagorean triple diophantine equation: $$ x^2 + y^2 = z^2, $$ where $x,y,z > 0$ are integers. I came across the ...
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How to appreciate Fermat's last theorem?

I am someone who is not a Maths major, these days (during the summer) I am attracted to Fermat's Last Theorem. I understand that there is no whole number solution to the equation $x^n + y^n = z^n$ for ...
8
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3answers
192 views

For which $n$ can $(a, nb, c)$ and $(b, c, d)$ be Pythagorean triples?

Fermat proved that if $(a, b, c)$ is a Pythagorean triple, then $(b, c, d)$ cannot be a Pythagorean triple. Suppose $(a, nb, c)$ form a Pythagorean triple. Can $(b, c, d)$ be a Pythagorean triple? ...
8
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1answer
153 views

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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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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5answers
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Can every perfect square exist as the sum or difference of two perfect squares?

I believe this is trivial and I'm over-complicating it. But can every squared integer be expressed as the sum of two squared integers OR the difference of two squared integers? And is there a proof ...
7
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3answers
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Is there an existing theory of Pythagorean triples for numbers of the form $p+q\sqrt r$ rather than integers?

Before Christmas I was teaching a class about surds. They were able to simplify, add, multiply etc. To give them one application of this, I wanted to give them some triangles that they would have to ...
7
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1answer
221 views

Is there a general formula for three Pythagorean Triangles which share an area?

The basic formula for generating a Pythagorean triangle $A^2 + B^2 = C^2$ is, $A = M^2 - N^2;\quad B = 2MN ;\quad C = M^2 + N^2$ And Wolfram Alpha gave me a solution (credited to an Enrique Zeleny)...
7
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1answer
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The Boolean Pythagorean triples problem, a $200$-terabyte proof, and $d=163$

I came across this interesting math article, "Computer cracks 200-terabyte maths proof" where one phrase caught my attention and I quote, "... all triples could be multi-coloured in integers up to $...
6
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2answers
915 views

Existence of a Pythagorean Triple with one side given.

I am curious about the answer to the question: Does there exists a pythagorean triple with $n$ as one of the sides for all $n\geq 3$ ?. Your answers and comments will mean a lot.
6
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3answers
391 views

Pythagorean Triple where $a=b$?

I cannot find even a single webpage mentioning this topic. I'm a programmer and I'm looking for a 45-45-90 triangle where all of the sides are whole numbers. In the video I am watching, they say to ...
6
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1answer
394 views

If $a^2 + p^2 = b^2$ then $2(a+p+1)$ is a perfect square

We are given $$ a^2 + p^2 = b^2 $$ where $a,b\in\mathbb{Z}$ and $p$ is prime. We are to show that $$2(a+p+1)$$ is a perfect square. Is there any elegant ways to go about this problem? Struggling to ...
6
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4answers
560 views

Prime numbers yield from Pythagoras triples

Pythagoras theorem $$a^2+b^2=c^2$$ we got $$P_{prime}(a,b)={a^4+b^4+(a+b)^4\over a^2+b^2+(a+b)^2}$$ Where $(a,b,c)$ are Pythagoras theorem triples, this function $P_{prime}(a,b)$ always produce a ...
6
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3answers
112 views

Let $(a, b, c)$ be a Pythagorean triple. Prove that $\left(\dfrac{􀀀c}{a}+\dfrac{􀀀c}{b}\right)^2$ is greater than 8 and never an integer.

Let $(a, b, c)$ be a Pythagorean triple, i.e. a triplet of positive integers with $a^2 + b^2 = c^2$. a) Prove that $$\left(\dfrac{􀀀c}{a}+\dfrac{􀀀c}{b}\right)^2 > 8$$ b) Prove that there ...
6
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4answers
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Pythagorean Triples : Is every positive integer $\gt$ $2$ part of at least one Pythagorean triple?

I was doing some basic number theory problems from Rosen and came across this problem: Show that every positive integer $\gt$ $2$ is part of at least one ...
6
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2answers
180 views

Is there $a,b,c,d\in \mathbb N$ so that $a^2+b^2=c^2$, $b^2+c^2=d^2$? [duplicate]

Question: Are there $a,b,c,d \in \mathbb N$ such that $$a^2 + b^2 = c^2 \ \ \text{and} \ \ b^2 + c^2 = d^2$$ I'm a bit lost here.
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 ...
6
votes
1answer
496 views

If $2xy$ is a perfect square, then $x^2+y^2$ cannot be

I ask a question that is probably never proved a conjecture: if $x$ and $y$ are two natural numbers $> 1$ such that $2xy = N^2$ ( double their product is a perfect square ), x and y cannot be part ...
6
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1answer
180 views

Cuboid nearest to a cube

Cuboid nearest to a cube. While answering this question, euler bricks: way to calculate them? I noticed one result was not too far from cube shaped, and wondered if there was a more cubic cuboid. $$...