No Quadruple of Consecutive Sums of Squares Exists

Theorem
It is not possible for a quadruple of consecutive positive integers each of which is the sum of two squares.

Proof
$4$ consecutive positive integers will be in the forms:

in some order.

But from Sum of Two Squares not Congruent to 3 modulo 4, $n_3$ cannot be the sum of two squares.

The result follows.