Writing 50 as a sum of two squares

Text

In how many ways can $50$ be written as a sum of two squares of positive integers $a^2+b^2$ with $1\le a\le b$? Find all the representations.

Solution

Try $a=1,2,\dots$ while $a^2\le 25$ (i.e. $a\le b$):

• $a=1$: $50-1=49=7^2$, so $1^2+7^2$ ✓
• $a=2$: $50-4=46$, not a square
• $a=3$: $50-9=41$, not a square
• $a=4$: $50-16=34$, not a square
• $a=5$: $50-25=25=5^2$, so $5^2+5^2$ ✓

There are therefore $\boxed{2}$ representations: $\ 50 = 1^2+7^2 = 5^2+5^2$.