An Olympiad Problem

Show that the equation has infinite integer solution : $x^2 + y^2 + z^2 = (x - y)(y - z)(z - x)$ .

#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

Interesting question. What have you tried?

Calvin Lin Staff - 5 years, 9 months ago

first of all, i want to thank you as you edited my que.. I tried but couldn't get any pattern.. And nobody is giving solution.

Dev Sharma - 5 years, 9 months ago

Try listing out a few small solutions. E.g. $(0, 0, 0), (-1, 0, 1), ( -1, 1, 2), (-2, -1, 1 )$ . Are there any others?

Calvin Lin Staff - 5 years, 9 months ago

I started the question by applying the substitution $y=x+a, z=x+a+b$ . We will restrict $a,b$ to be positive integers. This will give us a quadratic in $x$ which I omit here. My first thought, a lucky one, was to look at the case $a=b$ : $3x^2+6ax+5a^2-2a^3$ . After the quadratic we have:

$x=-a\pm \frac{a\sqrt {6(a-1)}}{3}$ .

We want $x$ to be an integer, so $a-1=6n^2$ for some integer $n$ .

Hence $x=(6n^2+1)(-1\pm 2n)$ , $y=(6n^2+1)(-1\pm 2n)+6n^2+1, z=(6n^2+1)(-1\pm 2n)+2(6n^2+1)$ . $n$ is an integer so we can generate an infinite number of integer solutions.

In all the proving infinite solution problems I've encountered in the past, a majority of them required looking at special cases; this should be a natural approach especially for an olympiad problem. The most complicated one I've dealt with involved special cases and pell's equation, can't remember the exact problem tho.

Xuming Liang - 5 years, 9 months ago

Thanks.

but why did you make y = x + a ....? and why only one case a = b??

Dev Sharma - 5 years, 9 months ago

the substitution is motivated by the product of differences on the right. I only examine the $a=b$ because that's all it suffices to show the infinitude of the solutions.

Xuming Liang - 5 years, 9 months 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}`	$\boxed{123}$