# STEP I-2006-1 and mental arithmetic

STEP I/2006/1 – Find the integer, n, that satisfies $n^{2} < 33\,127 < (n + 1)^2$. Find also a small integer m such that $(n + m)^2 - 33\,127$ is a perfect square. Hence express 33,127 in the form pq, where p and q are integers greater than 1.

By considering the possible factorisations of 33,127, show that there are exactly two values of m for which $(n + m)^2 - 33\,127$ is a perfect square, and find the other value.