The number of non-square numbers between 2022 and 2032 is:

Concept

For two consecutive perfect squares, n2 and (n + 1)2, every integer that lies strictly between them is automatically a non-square number — no other integer's square can fall strictly between two consecutive perfect squares. Expanding the identity (n + 1)2 − n2 = 2n + 1 shows that exactly 2n + 1 integers lie in the closed range from n2 + 1 to (n + 1)2 inclusive. One of those integers is the perfect square (n + 1)2 itself, so the count of non-square numbers strictly between n2 and (n + 1)2 is (2n + 1) − 1 = 2n.

Application

Here the two given squares are 2022 and 2032, so n = 202 and n + 1 = 203.

1. Write the expansion of the difference of the two given squares: (n + 1)2 − n2 = n2 + 2n + 1 − n2 = 2n + 1.

2. Substitute n = 202: 2032 − 2022 = 2(202) + 1 = 405.

3. This 405 is the count of integers in the closed range from 2022 + 1 to 2032 inclusive (405 consecutive integers).

4. Exactly one of those 405 integers, namely 2032 itself, is a perfect square; every other integer in that closed range is strictly between 2022 and 2032 and is therefore non-square (since 2022 and 2032 are consecutive perfect squares, no perfect square can lie strictly in between them).

5. So the number of non-square integers strictly between 2022 and 2032 is 405 − 1 = 404.

Cross-check

Compute the two squares directly: 2022 = 40804 and 2032 = 41209. The integers strictly between them run from 40805 to 41208, which is 41208 − 40805 + 1 = 404 integers — matching the result obtained from the general identity 2n = 2(202) = 404, and confirming that none of these 404 integers is a perfect square, since the only two perfect squares near this range are the endpoints 2022 and 2032 themselves.

Result

The number of non-square numbers between 2022 and 2032 is 404.