# 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
4 years, 1 month ago

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} $$\boxed{123}$$

Sort by:

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.

- 4 years, 1 month ago

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 - 4 years, 1 month ago

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

- 4 years, 1 month ago

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.

- 4 years, 1 month ago

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

- 4 years, 1 month ago

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

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

- 4 years, 1 month ago

- 4 years, 1 month ago

- 4 years, 1 month ago

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

- 4 years, 1 month ago

But 32 isn't a perfect square.

- 4 years, 1 month ago

Yeah...

- 4 years, 1 month ago

but $$17+8$$ is

- 4 years, 1 month ago

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

- 4 years, 1 month ago

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.

- 4 years, 1 month ago

Yes I "thnik" you can. :D

- 4 years, 1 month ago