Help needed friends!

$x,y \in Z^{+}\\ (x+y)^2+3x+y=1994\\ y=?$

#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

To make this problem much simpler, let us make a substitution $x = \dfrac{u+v}{2}$ and $y=\dfrac{u-v}{2}$ .

So, the given equation converts into

$u^2 + 2u + \left( v- 1994 \right) = 0$

Step-1: Let us first find the lower bound and upper bound of $v$ . It is clear that $u = x+y$ and $v = x-y$ . So, $u > v$ using this fact we get

$\begin{aligned} 1994 = u^2 + 2u +v > v^2 + 3v \\ v^2 + 3v - 1994 < 0 \end{aligned}$

Since $x$ and $y$ are integers and so $v$ is an integer. Then from the above inequality, we get $v \in \Big[ -92, 86 \Big]$ .

Step-2: Since from the equation $u^2 + 2u + \left( v- 1994 \right) = 0$ . We get, $u= -1 \pm \sqrt{1995-v}$ . But since $x>0$ and $y>0$ it follows that $u>0$ . So, $u= -1 + \sqrt{1995-v}$ .

Now for positive integral value of $u$ , $1995-v$ should be a perfect square. Now from the step-1 $1995-v \in \Big[ 1909, 2087 \Big]$ . There are only two values of $1995 -v$ , they are $2025$ and $1936$ .

They yield the following pairs of $\left(u,v \right)$ . They are $\{ \left(43,59 \right) , \left(44, -30 \right) \}$ . But since $u>v$ , we get only pair of $\left( u,v\right)$ i.e. $\left( 44, -30 \right)$ which yields $x = 7$ and $y= 37$ .

So, therefore only one solution of $\left( x , y \right)$ exists i.e. $\left( 7 , 37 \right)$ .

Surya Prakash - 5 years, 8 months ago

Thanx for the solution!An awesome one!