Need solution

Find the positive integers $n$ for which $n+9,16n+9$ and $27n+9$ are all perfect squares.

#NumberTheory

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.

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

Math	Appears 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

Let $n+9=x^2$ $16n+9=y^2$ and $27n+9=z^2$ now multiply the first equation with 16 and subtract it with the second. You'll be left with $16x^2-y^2=135$ Similarily with the first and third mulitiply with 27, you'll be left with $27x^2-z^2=234$ . For the last multiplication step multiply the second with 27 and and third with 16 you'll be left with $27y^2-16z^2=99$ now $27y^2-16z^2+16x^2-y^2=234=27x^2-z^2$ By arranging that we get $26y^2=15z^2+11x^2$ Now the obvious solution is $x=y=z=1$ which leaves us with a contradiction that no such integers n exist. However I am not sure that there are no more solutions to this equation(I haven't come across that field of study yet!) Good luck I hope I helped

Dragan Marković - 5 years, 1 month ago

Nice sol.+1..how did u think of that?

Rishabh Tiwari - 5 years, 1 month ago

Well actually with elimination. I have 2 appoaches to such problems. Either factorization or eliminating the unknowns. Factorization failed horribly i got nothing from it so i tried this way and ended up with the final equation. So i guess the factroization didn't fail that horribly? And this method was way shorter to write ;)

Dragan Marković - 5 years, 1 month ago

Hey! See this, $n=280$ satisfies.

Akshat Sharda - 5 years ago

It is the solution to my equation as well!!! I don't know how to get 17 67 and 87 though we haven't done that in school yet. Maybe someone like Calvin Lin could help us with that?

Dragan Marković - 5 years ago

Wow did some more analysis. If we can prove that $7y=5z+2x 《》25y=30z-55x$ we would get 280 for n. This is intriguing 7=5+2 and 25=55-30!!!!

Dragan Marković - 5 years ago

Math	Appears 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}`	$\boxed{123}$