Questions tagged [pythagorean-triples]
Questions about Pythagorean triples, positive integer solutions to $a^2 + b^2 = c^2$.
0
votes
3answers
69 views
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 ...
-1
votes
2answers
42 views
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 ...
0
votes
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=...
2
votes
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:
$$x=r^2-s^2$$
$$...
-1
votes
2answers
38 views
We can't exactly draw a line of length square root of 2 but can be constructed using Pythagoras theorem in an isosceles right angle triangle?
We can't exactly draw a line of length square root of 2 but in an isosceles right angle triangle of sides 1 unit each, the length of hypotenuse will be the square root of 2. Now does it mean we can ...
0
votes
1answer
34 views
Euler Brick calculation
How would you go about calculating Euler bricks from a list of primitive Pythagorean triples. I've tried to find an answer to this online but can't find anything which gives me list of Euler bricks ...
2
votes
2answers
89 views
Find all Pythagorean triples $x^2+y^2=z^2$ where $x=21$
Consider the following theorem:
If $(x,y,z)$ are the lengths of a Primitive Pythagorean triangle, then $$x = r^2-s^2$$ $$y = 2rs$$ $$z = r^2+z^2$$ where $\gcd(r,s) = 1$ and $r,s$ are of opposite
...
0
votes
3answers
95 views
Pythagorean like Diophantine Equation
I am trying to solve this problem.
http://www.javaist.com/rosecode/problem-527-1-2-3-type-Pythagorean-triangles-askyear-2018
Here we have to find all positive integral solution of $a^2+2b^2=3c^2$ ...
7
votes
2answers
53 views
Prove that if $a, b, c \in \mathbb{Z^+}$ and $a^2+b^2=c^2$ then ${1\over2}(c-a)(c-b)$ is a perfect square.
Prove that if $a, b, c \in \mathbb{Z^+}$ and $a^2+b^2=c^2$ then ${1\over2}(c-a)(c-b)$ is a perfect square.
I have tried to solve this question and did pretty well until I reached the end, so I was ...
0
votes
3answers
47 views
Pythagorean triple
Show that neither $1$ not $2$ can appear in any Pythagorean triple, but that every integer $k\geq3$ can appear.
Prove that for each integer $k$ there are only finitely many Pythagorean triple ...
0
votes
1answer
45 views
Another proof for an infinite number of Pythagorean triples
I’m not sure if this has been mentioned before (and I truly apologize if someone thought about it already) , but I tried to adopt a geometrical approach for the proof for an infinite number of ...
1
vote
2answers
84 views
Did Babylonians know the Pythagorean theorem before his time?
On old tablets the Babylonians were able to work out the digits to the square root of two from the hypotenuse of a $45^\circ-45^\circ-90^\circ$ triangle.
How could they have figured this out without ...
3
votes
2answers
42 views
Basic questions about pythagorean triples and “n-lets”
I've had some difficulties finding answer to the two following questions:
1) Given one of natural numbers $a,b$ where $b$ is even and $a^2+b^2=c^2$ is there only one such a pythagorean triple?
2)How ...
1
vote
1answer
32 views
Integer solutions to AGM iteration
Any integer solution to $a^2+b^2=c^2$ also provides an integer solution $x=c$, $y=a$, $z=c+b$, $w=c-b$ to $$agm(x,y)=agm(z,w)$$ where $agm$ denotes Gauss' arithmetic-geometric mean. Are there other ...
0
votes
1answer
57 views
Prove for every $m, n$, $(m^2 - n^2, 2mn, m^2 + n^2)$ can make pythagorean triple
I tried researching more about this because it seems to be a common topic, but I don't know how to approach this problem. Do I have to somehow arrange those 3 terms into $a^2 + b^2 = c^2$?
2
votes
3answers
394 views
For any primitive pythagorean triple $(a,b,c)$ either $a$ or $b$ must be a multiple of $3$ [duplicate]
I'm reading "Friendly Introduction to Number Theory". Now I'm working on Primitive Pythagorean Triples Exercises 2.1 (a) on P18.
We showed that in any primitive Pythagorean triple $(a, b, c)$, ...
1
vote
2answers
48 views
Generating Pythagorean triples
I'm asked to generate Pythagorean triples from the polynomial identity:
$$(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 ...
2
votes
1answer
127 views
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 ...
-1
votes
1answer
39 views
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:...
1
vote
1answer
43 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 - ...
-1
votes
1answer
51 views
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}$
3
votes
1answer
111 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 ...
1
vote
2answers
96 views
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:
...
1
vote
0answers
40 views
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$$
I ...
2
votes
0answers
40 views
Matrix valued Pythagorean Triples
Consider any nxn matrices A, B and C such that
A^2 + B^2 = C^2
Then the matrix triple (A,B,C) is called a Matrix valued Pythagorean Triple.
I have observed that any nxn matrix M and N such that MN=NM,...
10
votes
2answers
149 views
Solve $ \binom{a}{2} + \binom{b}{2} = \binom{c}{2} $ with $a,b,c \in \mathbb{Z}$
I am trying to solve the Diophantine equation:
$$ \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) + ...
2
votes
2answers
82 views
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
$\gcd(a,d,e)=1$
$a^2+d^2=e^2$
?
0
votes
1answer
145 views
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 ...
0
votes
2answers
38 views
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}...
1
vote
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 [...
1
vote
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 ...
2
votes
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,...
12
votes
4answers
430 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 ...
3
votes
0answers
36 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+\...
5
votes
1answer
72 views
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 ...
10
votes
2answers
121 views
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$, ...
0
votes
1answer
55 views
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.
1
vote
2answers
45 views
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 ...
2
votes
0answers
39 views
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$, ...
10
votes
3answers
245 views
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 ...
-3
votes
2answers
169 views
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....
0
votes
1answer
23 views
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 $$ is divisible by $c$.
Any hints??
0
votes
3answers
56 views
In a unit circle, what allows the same trigonometric rules to be applied past the first quadrant?
I had previously asked this question and was asked to learn more about the unit circle, which I've done. I now have further questions:
When the concept of trigonometric ratios for acute angles is ...
2
votes
2answers
63 views
No primitive pythagorean triple $(x,y,z)$ with $z\equiv -1 \pmod 4$
Show that there are no primitive pythagorean triple $(x,y,z)$ with $z\equiv -1 \pmod 4$.
I once have proven that, for all integers $a,b$, we have that $a^2 + b^2$ is congruent to $0$, or $1$, or $2$ ...
0
votes
0answers
40 views
Are there any (more) consecutive numbers that are functional in the Pythagorean Theorem?
The numbers three, four, and five can be implanted in the Pythagorean process (3=a, 4=b, 5=c) to equal a correct right triangle. But my question is, are there any more directly consecutive numbers ...
0
votes
2answers
56 views
Given primitive Pythagorean triple $(a,b,c)$ with $a<b<c$, if $a$ is odd, then $a^2 = b+c$
I've noticed that, for any pythagorean triple $(a,b,c)$ arranged least to greatest, if they have no common divisor (i.e., if the triple is primitive) and $a$ is odd, then following holds:
$$a^2=b+c$$
...
0
votes
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 ...
5
votes
2answers
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
votes
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 ...
1
vote
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 ...
