No Quadruple of Consecutive Sums of Squares Exists

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


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

\(\ds n_0\) \(\equiv\) \(\ds 0\) \(\ds \pmod 4\)
\(\ds n_1\) \(\equiv\) \(\ds 1\) \(\ds \pmod 4\)
\(\ds n_2\) \(\equiv\) \(\ds 2\) \(\ds \pmod 4\)
\(\ds n_3\) \(\equiv\) \(\ds 3\) \(\ds \pmod 4\)

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.