Can You Discover the Integer Pair That Satisfies This Unique Sum Equation?

[anemone]

Here is this week's POTW:

-----

Find a pair $(a,\,b)$ of positive integers $a<b$ such that

$37^2+46^2+49^2-20^2-17^2=a^2+b^2$

-----

Remember to read the https://mathhelpboards.com/showthread.php?772-Problem-of-the-Week-%28POTW%29-Procedure-and-Guidelines to find out how to https://mathhelpboards.com/forms.php?do=form&fid=2!

 

[anemone]

Congratulations to the following members for their correct solution:):

1. castor28
2. kaliprasad

Solution from castor28:

Spoiler

We may write:
\begin{align*}
S &= 37^2+46^2+49^2-20^2-17^2\\
&= 37^2 +(46^2 - 20^2) + (49^2 - 17^2)\\
&= 37^2 + (66\times26) + (66\times32)\\
&= 37^2 + 66\times58 \\
&= 37^2 + 66\times(66-8)\\
&= 37^2 + 66^2 - 8\times66\qquad [1]
\end{align*}
We notice that $8+66=2\times37$; this suggests to write $37^2 - 29^2 = 8\times66$; substitution in [1] gives $S = 29^2+66^2$, i.e., $a=29$, $b=66$.