No Quadruple of Consecutive Sums of Squares Exists
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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:
\(\ds n_0\) | \(\equiv\) | \(\ds 0\) | \(\ds \pmod 4\) | |||||||||||
\(\ds n_1\) | \(\equiv\) | \(\ds 1\) | \(\ds \pmod 4\) | |||||||||||
\(\ds n_2\) | \(\equiv\) | \(\ds 2\) | \(\ds \pmod 4\) | |||||||||||
\(\ds n_3\) | \(\equiv\) | \(\ds 3\) | \(\ds \pmod 4\) |
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.
$\blacksquare$
Sources
- 1986: David Wells: Curious and Interesting Numbers ... (previous) ... (next): $232$
- 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $232$