Stack Exchange Network

Stack Exchange network consists of 175 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

Visit Stack Exchange

Questions tagged [pythagorean-triples]

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

8
votes
4answers
3k views

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 ...
16
votes
2answers
1k views

$x^2+y^2=z^n$: 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^...
16
votes
7answers
5k views
3
votes
2answers
223 views

On $p^2 + nq^2 = z^2,\;p^2 - nq^2 = t^2$ and the “congruent number problem”

(Much revised for brevity.) An integer $n$ is a congruent number if there are rationals $a,b,c$ such that, $$a^2+b^2 = c^2\\ \tfrac{1}{2}ab = n$$ or, alternatively, the elliptic curve, $$y = x^3-n^...
1
vote
7answers
6k views

Formulas for calculating pythagorean triples

I'm looking for formulas or methods to find pythagorean triples. I only know one formula for calculating a pythagorean triple and that is euclid's which is: $$\begin{align} &a = m^2-n^2 \\ &b ...
1
vote
2answers
2k views

Primitive Pythagorean triple divisible by 3

Prove that for any primitive Pythagorean triple (a, b, c), exactly one of a and b must be a multiple of 3, and c cannot be a multiple of 3. My attempt: Let a and b be relatively prime positive ...
7
votes
1answer
955 views

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 $...
8
votes
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? ...
9
votes
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.$...
5
votes
3answers
203 views

When is $5n^2+14n+1$ a perfect square?

This specific quadratic came up as part of a puzzle, but the context isn't really important. I just need to find all positive integers $n$, where $5n^2+14n+1$ is a perfect square. Unfortunately I'm ...
4
votes
1answer
295 views

Finding two non-congruent right-angle triangles

The map $g: B \to A, \ (x,y) \mapsto \left(\dfrac {x^2 - 25} y, \dfrac {10x} y, \dfrac {x^2 + 25} y \right)$ is a bijection where $A = \{ (a,b,c) \in \Bbb Q ^3 : a^2 + b^2 = c^2, \ ab = 10 \}$ and $B =...
131
votes
1answer
4k views

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 ...
5
votes
1answer
237 views

number of primitive Pythagorean triangles whose hypotenuses do not exceed n?

i just read "mathematical constants" book; it said that Lehmer proved the following theorem in 1900 where P_h(n) , P_p(n) is number of primitive Pythagorean triangles whose hypotenuses and perimeter ...
6
votes
2answers
292 views

Are there any positive integers $a, b, c, d$ such that both $(a, b, c)$ and $(b, c, d)$ are Pythagorean triples?

Pythagorean triple is a triple of integers $(a, b, c)$ such that $a^2+b^2=c^2$. Is there any Pythagorean triple such that, not only $a^2+b^2$, but also $b^2+c^2$ is a square number? If not, how to ...
6
votes
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.
3
votes
4answers
558 views

the converse of Pythagoras Theorem

Is it possible to prove the converse of the Pythagoras theorem with out a geometric proof?.That is from $a^2+b^2=c^2$ to $\Theta=90^\circ$
3
votes
2answers
486 views

Primitive Pythagorean triple generator

I was wondering how to prove the following fact about primitive Pythagorean triples: Let $(z,u,w)$ be a primitive Pythagorean triple. Then there exist relatively prime positive integers $a,b$ of ...
1
vote
4answers
566 views

Show that for any prime numbers $p,q,r$, one has $p^2+q^2 \ne r^2$. [closed]

Not sure how to start. Prove by induction? Please help!
1
vote
1answer
108 views

Geometric interpretation of $x^3+y^3+z^3=k^3$

$a^2+b^2=c^2$ is related to right triangles - how the sums of the squares of the legs equals the square of the hypotenuse. What about the following sum of cubes? $$x^3+y^3+z^3=k^3$$ Is there a ...
0
votes
2answers
76 views

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-...
0
votes
1answer
2k views

How to Prove Pythagorean Triple Formula

I'm having a hard time finding a proof for how they derived the Pythagorean triple formula. It's hard to find the proof online and When I do find it, it's hard to understand. $a = p^2 − q^2 , b = ...
11
votes
2answers
1k views

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 ...
12
votes
0answers
436 views

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 ...
3
votes
2answers
158 views

Triples of positive integers $a,b,c$ with rational $\sqrt{\frac{c-a}{c+b}},\sqrt{\frac{c+a}{c-b}}$

While working on a physics problem, I came up with a certain question in number theory: For positive integers $c>b>a$, can $\dfrac{c-a}{c+b}$ and $\dfrac{c+a}{c-b}$ both be rational squares? ...
23
votes
5answers
3k views

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 ...
3
votes
2answers
222 views

A pair of continued fractions that are algebraic numbers and related to $a^2+b^2=c^m$

Similar to the cfracs in this post, define the two complementary continued fractions, $$x=\cfrac{-(m+1)}{km\color{blue}+\cfrac{(-1)(2m+1)} {3km\color{blue}+\cfrac{(m-1)(3m+1)}{5km\color{blue} +\cfrac{...
12
votes
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 ...
9
votes
3answers
363 views

Three pythagorean triples

Are there any solutions for $a, b, c$ such that: $$a, b, c \in \Bbb N_1$$ $$\sqrt{a^2+(b+c)^2} \in \Bbb N_1$$ $$\sqrt{b^2+(a+c)^2} \in \Bbb N_1$$ $$\sqrt{c^2+(a+b)^2} \in \Bbb N_1$$
7
votes
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)...
5
votes
1answer
277 views

a continued fraction related to pythagoras theorem $a^2+b^2=c^2$

For our purpose,let $a,b,c$ and $x\gt2$ be natural numbers such that the positive integers $a,b$ and $c$ form a special pythagorean triple $(a,b,c)$,then it is conjectured that the following is true $...
3
votes
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, ...
3
votes
1answer
149 views

Special Euler bricks and $x^2(y^2-1)^2+y^2(x^2-1)^2=z^2$

Define, $$P_1 := a^2+b^2\\ P_2 := a^2+c^2\\ P_3 := b^2+c^2$$ Let, $$a,b,c = 2xy,\;x(y^2-1),\;y(x^2-1)$$ and $P_1,P_2$ become squares. If we wish to make $P_3$ a square as well, then, $$P_3:=x^2(...
2
votes
0answers
114 views

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 ...
9
votes
2answers
486 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, \...
7
votes
5answers
4k views

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 ...
6
votes
2answers
150 views

Solve for Rationals $p,q,r$ Satisfying $\frac{2p}{1+2p-p^2}+\frac{2q}{1+2q-q^2}+\frac{2r}{1+2r-r^2}=1$.

Find all rational solutions $(p,q,r)$ to the Diophantine equation $$\frac{2p}{1+2p-p^2}+\frac{2q}{1+2q-q^2}+\frac{2r}{1+2r-r^2}=1\,.$$ At least, determine an infinite family of $(p,q,r)\in\mathbb{Q}^3$...
6
votes
2answers
135 views

How many integer-sided right triangles are there whose sides are combinations?

How many integer-sided right triangles exist whose sides are combinations of the form $\displaystyle \binom{x}{2},\displaystyle \binom{y}{2},\displaystyle \binom{z}{2}$? Attempt: This seems like a ...
6
votes
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. $$...
6
votes
1answer
492 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 ...
5
votes
4answers
714 views

How to descend within the “Tree of primitive Pythagorean triples”?

It is well-known that the set of all primitive Pythagorean triples has the structure of an infinite ternary rooted tree. What is the exact algorithm (i.e., formula, or possibly set of three formulas) ...
4
votes
1answer
120 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 ...
3
votes
1answer
192 views

On $119^2+120^2=13^4\,$ and $p=239$

We are familiar with, $$\frac\pi4=4\arctan\tfrac15-\arctan\tfrac1{239}$$Let $p=a+b=239$ and $(a,b,c,d)=(120,119,13,2).\,$ Some years back, I observed this rather long list of Diophantine relations and ...
1
vote
2answers
1k views

Pythagorean Triple divisible by $5$

Show that, if x, y and z are integers such that $x^2+y^2 = z^2$,then at least one of $x,y,z$ is divisible by $5$. I was able to show that at least one of $x$ or $y$ is divisible by $2$. Can someone ...
1
vote
3answers
3k views

proof: primitive pythagorean triple, a or b has to be divisible by 3

I'm reading "A friendly introduction to number theory" and I'm stuck in this exercise, I'm mentioning this because what I need is a basic answer, all I know about primitive pythagorean triplets is ...
1
vote
3answers
1k views

Find all solutions to the equation $x^2 + 3y^2 = z^2$

Find all positive integer solutions to the equation $x^2 + 3y^2 = z^2$ So here's what I've done thus far: I know that if a solution exists, then there's a solution where (x,y,z) = 1, because if there ...
3
votes
2answers
151 views

solutions of $a^2+b^2=c^2$

I am trying to figure the following out. If you have $a^2+b^2=c^2$ and let $x=a/c$ and $y=b/c$ how can you show that $x=\frac{m^2-n^2}{m^2+n^2}$ and $y=\frac{2mn}{m^2+n^2}$ for some relatively ...
3
votes
1answer
2k views

Are there infinitely many pythagorean triples?

I believe these questions are all asking different things, but: Are there infinitely many (integer) solutions to the pythagorean theorem? Is every positive integer part of a solution to the ...
2
votes
3answers
848 views

Pythagorean triples with the same c value

$a^2 + b^2 = c^2$ There are, Primitive Pythagorean Triples, that share the same c value. For example, $63^2 + 16^2 = 65^2$ and $33 ^2 + 56^2 = 65^2$. I have been trying to figure out why the ...
2
votes
1answer
149 views

Find pairs of side integers for a given hypothenuse number so it is Pythagorean Triple

I am trying to find all pairs of side integers (a, b) for a given hypothenuse number n so that (a, b, n) is a Pythagorean triple, i.e.,$ a^2 + b^2 = n^2$ The approach i am using is Sorting the ...
2
votes
1answer
827 views

Vectors: Using Pythagoras's theorem for magnitude in the 4th dimension

For a simple x and y plane (2 dimensional), to find the distance between two points we would use the formula $$ a^2 +b^2 = c^2 $$ For a slightly more complicated plane; x,y and z (3 dimensional), ...