# Questions tagged [pythagorean-triples]

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

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### Tiling potential of the primitive triplets

What is the maximum percentage of an infinite plane that could be tiled with primitive Pythagorean triangles, if every triangle may be used at most once? My intuitive guess is that it can’t reach 100%,...
• 29
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### Pythagorean triangles with non rectilinear [closed]

Does there exist a Pythagorean triple whose corner is non-rectilinear with axes, yet all three vertices are on integer coordinates? The question was closed due to "lack of context". The ...
80 views

### Imaginary Units in the Pythagorean Theorem

Are there any exceptions to the Pythagorean Theorem? I ask this after seeing the triangle with sides i, $1$, and $0$. Sure, this triangle checks out with the Pythagorean Theorem ($i^2+1^2=0^2$), but ...
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### Distribution of primes in primitive Pythagorean triples

My Observation: I've observed a pattern where for every pair of twin primes ($p$, $p+2$), there appears to be at least one primitive Pythagorean triple ($a$, $b$, $c$) such that one of the twin primes ...
384 views

### Integer Right Triangle with Repdigit Area

This question was inspired by a tweet by Cliff Pickover. I'm seeking a right triangle with integer sides, and whose area is a repdigit number greater than $666666$. I know that Pythagorean Triples can ...
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### Finding the Pythagorean triples for a given small side

I wish to find all Pythagorean triples containing a given number as one of it's small sides. Take a right angled triangle $a,b,c$ small side $a$ and hypotenuse $c$. Let $N=c-b$ Substituting into ...
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• 6,416
1 vote
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### Why does the term being squared in the Pythagorean triplet make no difference to the whole triplet?

I was recently preparing for the SOF IMO Level 2 exam. While practising for the exam, I encountered this question in my textbook about the chapter Square and Square Roots which also mentions the ...
1 vote
74 views

### Pythagoras theorem twice [duplicate]

Is there any pair of positive integers,$a,b,c,d$,that $a^2+b^2=c^2$ and $b^2+c^2=d^2$ holds true? Thanks!
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### Four rational Pythagorean triples in a square

Suppose we have a unit square $ABCD$. If we have a primitive Pythagorean triple $(a, b, c)$, let $CD = 1$ and hence $ED = a/b$. It also follows that $EC$ is rational of course. In fact, using the ...
• 113
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### How to find the side lengths of a triangle in the incircle of a Pythagorean triple

Given a Pythagorean triple with an incircle, how do I find the sides of a triangle connecting the triple's tangents to that incircle. In the diagram below, I know how to find side $\,i\,$ but not the ...
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### Radii of touching spheres centered on Pythagorean triple vertices.

There are similar questions and answers out there but other answers I have seen seem unnecessarily complicated. I came up with what I think is a simple approach some time ago but I'm now doubting my ...
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### pythagorean triple factorization

Let $z>y>x$ primitive pythagorean triple. Show that $z$ has a prime factorization to primes of the form $4k+1$ I tried using the fact that there exist $s,t$ such that $s$ is not congruent to $t$ ...
• 211
1 vote
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### Constant Hiding Behind Particular Pythagorean Triples [duplicate]

The question pertains to an apparent constant appearing behind Pythagorean Triples whose "legs" have a difference of 1. To start, here are the first 5 triples that satisfy the above ...
• 183
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### Fairly simple problem: three integers, sums, and Pythagorean triples

So, I randomly thought about this simple problem, and very sadly couldn't make much progress in answering this. Does there exist three integers $a,b,c\neq0$ where $a^2+b^2$, $a^2+c^2$, and $b^2+c^2$ ...
129 views

### Rotation matrix and Pythagorean theorem relation

From Wikipedia, the trace of the product of $R_z, R_y$, and $R_x$ is equal to $1 + 2\cos(\theta)$. Solving for $\theta$, we get $\theta$ = $\arccos\left(\frac{\text{trace} - 1}{2}\right)$. What is ...
87 views

### Is the set of “Pythagorean” square roots dense in $\left[0, \infty\right)$? [closed]

I was thinking about proofs of the fact that square roots exist and ended up wondering if the set of numbers of the form $\sqrt{\dfrac{a^2}{b^2}+ \dfrac{c^2}{d^2}}$ with $a, b, c, d \in\Bbb N$ is a ...
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### If $x^2+px-q=0$ and $x^2+px+q=0$ have integer roots for real $p$ and $q$, then $p^2=a^2+b^2$ for some integers $a$ and $b$

Two equations $x^2+px-q=0$ and $x^2+px+q=0$ have integer roots, where $p$ and $q$ are real numbers. Prove that for some integers $a$ and $b$, $p^2 = a^2 + b^2$. In other words, have $\{a,b,p\}$ be a ...
174 views

### What is the area of the shaded region if $AB=120$?

What is the area of the blue region if $AB=120$. I can only think in $\triangle OAB$ (O b being the low left point and the triangle is right) I can use Pythagoras which involves $120$, radius $r$ of ...
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### Can I infer the hypotenuse given only $a > b$?

Today, I came up with a problem. The problem is this: Let $a$, $b$ and $c$ be the sides of a right-angled triangle, such that $a > b$. The nature of $c$ is unknown. Can I infer the hypotenuse given ...
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