Sum of Consecutive Odd Numbers

Let the 4 consecutive odd numbers be $2k-3, 2k-1, 2k+1, 2k+3$. Sum $= 8k$. Sum / 10 $= \frac{8k}{10} = \frac{4k}{5} = p^2$ (a perfect square). Since $4k/5$ is a square, $k$ must be a multiple of 5, say $k = 5m^2$.

• If $m = 2 \implies k = 20 \implies$ sum $= 160 \implies 160/10 = 16 = 4^2$. The 4 numbers are 37, 39, 41, 43. 41 is one of these numbers.