Sunday Times Teaser 2990 – Squares and Cubes
by Howard Williams
Published January 12 2020 (link)
Jenny is pleased that she has found two whole numbers with a remarkable property. One of them is a single digit greater than zero while the other one has two digits. The remarkable thing is that the difference of their squares is a perfect cube and the difference of their cubes is a perfect square.
Her sister Sarah is not impressed, however. She has found two three-digit numbers for which the difference of their squares is also a perfect cube and the difference of their cubes is also a perfect square.
In ascending order, what are the four numbers?