# More Squares in Pythag Triples Imgur

Let $$a,b,c$$ be a Pythagorean Triple where $$a < b < c$$.

Problem 1: Find the condition that $a,b,c$ need to satisfy such that $b+c$ is a perfect square.

Problem 2: Find the condition that $a,b,c$ need to satisfy such that $a+c$ is a perfect square.

Problem 3: If you can, find a condition for $a,b,c$ for $a+b$ to be a perfect square. Note by Daniel Liu
7 years ago

This discussion board is a place to discuss our Daily Challenges and the math and science related to those challenges. Explanations are more than just a solution — they should explain the steps and thinking strategies that you used to obtain the solution. Comments should further the discussion of math and science.

When posting on Brilliant:

• Use the emojis to react to an explanation, whether you're congratulating a job well done , or just really confused .
• Ask specific questions about the challenge or the steps in somebody's explanation. Well-posed questions can add a lot to the discussion, but posting "I don't understand!" doesn't help anyone.
• Try to contribute something new to the discussion, whether it is an extension, generalization or other idea related to the challenge.
• Stay on topic — we're all here to learn more about math and science, not to hear about your favorite get-rich-quick scheme or current world events.

MarkdownAppears as
*italics* or _italics_ italics
**bold** or __bold__ bold
- bulleted- list
• bulleted
• list
1. numbered2. list
1. numbered
2. list
Note: you must add a full line of space before and after lists for them to show up correctly
paragraph 1paragraph 2

paragraph 1

paragraph 2

[example link](https://brilliant.org)example link
> This is a quote
This is a quote
    # I indented these lines
# 4 spaces, and now they show
# up as a code block.

print "hello world"
# I indented these lines
# 4 spaces, and now they show
# up as a code block.

print "hello world"
MathAppears as
Remember to wrap math in $$ ... $$ or $ ... $ to ensure proper formatting.
2 \times 3 $2 \times 3$
2^{34} $2^{34}$
a_{i-1} $a_{i-1}$
\frac{2}{3} $\frac{2}{3}$
\sqrt{2} $\sqrt{2}$
\sum_{i=1}^3 $\sum_{i=1}^3$
\sin \theta $\sin \theta$
\boxed{123} $\boxed{123}$

## Comments

Sort by:

Top Newest

Considering a, b, and c natural numbers:

For problem 1, a should be odd. For problem 2, a should be even. Can't figure out problem 3, as it seems impossible for natural numbers.

- 7 years ago

Log in to reply

(9,40,41) satisfies problem 3's conditions.

- 7 years ago

Log in to reply

The general Pythagorean triple solutions are $k.2ab, k(a^2-b^2)$ and $k(a^2+b^2)$.For #2 it suffices that k is a perfect square.For #1 2k needs to be a perfect square.

- 7 years ago

Log in to reply

But we require $a < b < c$. It is not true that $2xy < ( x^2 - y^2) < (x^2 + y^2 )$ (where I changed your notation from ab to xy).

Staff - 7 years ago

Log in to reply

But every pair of numbers is in the problems, so that doesn't change it very much.

- 7 years ago

Log in to reply

I'm not really doing any proofs. Here's my answer for #1. I'll do the rest when I have more time. Great note though! :D

Problem 1: If $b$ and $c$ are consecutive integers that add to a perfect odd square. For example, the general formula for an odd number $n=2x+1$ for $x>1$ is

$(2x+1, \lfloor\dfrac{(2x+1)^2}{2}\rfloor, \lceil\dfrac{(2x+1)^2}{2}\rceil)$

- 7 years ago

Log in to reply

What about (8,15,17)?

- 7 years ago

Log in to reply

What about it?

- 7 years ago

Log in to reply

It doesn't satisfy your requirement but it still satisfies #1...

- 7 years ago

Log in to reply

But 32 isn't a perfect square.

- 7 years ago

Log in to reply

Yeah...

- 7 years ago

Log in to reply

but $17+8$ is

- 7 years ago

Log in to reply

True, but we're talking about Problem 1, where $b+c$ is a perfect square.

- 7 years ago

Log in to reply

Oh oops, I accidentally thought you were talking about #2, don't know what happened. But I can still find other triplets that have $b+c$ be square and not satisfy your requirement I thnik.

- 7 years ago

Log in to reply

Yes I "thnik" you can. :D

- 7 years ago

Log in to reply

×

Problem Loading...

Note Loading...

Set Loading...