Hypatia’s Triumph

Hypatia was an Egyptian female mathematician born in A.D. 370. One problem she posed was the following:

Find a number that is the sum of two squares and whose square is also the sum of two squares.

Can Hypatia’s problem be solved? Is there more than one solution? If so, is there a pattern among the solutions? Justify the conclusions you reach.

(Source: Mathematics Teaching in the Middle School, Apr 1994)