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.

$4$ 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.

$\blacksquare$

1986: David Wells: Curious and Interesting Numbers ... (previous) ... (next): $232$
1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $232$