Question 28
a) There are infinitely many positive integer numbers $n$ such that there exists a square equal to the sum of the squares of $n$ consecutive positive integer numbers. (For instance, $2$ and $11$ are such: $5^2 = 3^2 + 4^2$ and $77^2 = 18^2 + 19^2 + \dots + 28^2$.)
b) If $n$ is a positive integer number which is not a perfect square and if $x_0$ is an integer number such that $x_0^2 + (x_0 + 1)^2 + \dots + (x_0 + n - 1)^2$ is a perfect square, then there are infinitely many positive integer numbers $x$ such that $x^2 + (x + 1)^2 + \dots + (x + n - 1)^2$ is a perfect square.
b) If $n$ is a positive integer number which is not a perfect square and if $x_0$ is an integer number such that $x_0^2 + (x_0 + 1)^2 + \dots + (x_0 + n - 1)^2$ is a perfect square, then there are infinitely many positive integer numbers $x$ such that $x^2 + (x + 1)^2 + \dots + (x + n - 1)^2$ is a perfect square.