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 $x = 857.$