1332.11 – Whole Number Radicals

From the given information, $x+43$ and $x - 16$ are perfect squares that differ by 59. That this is an odd number should remind students of Stella 1331.14 from this exercise set. In that problem, the key result is the identity

$$(n+1)^2 - n^2 = 2n + 1$$

which expresses the difference between two consecutive squares as a generic odd number. Thus to solve this problem we set $2n + 1 = 59$ to find that $n = 29$.

Therefore two squares that work are $29^2 = 841$ and $30^2 = 900.$ It follows that $x+43 = 900$ so one answer is $x = 857$.