Numbers whose Squares are Consecutive Odd or Even Integers Juxtaposed

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Theorem

Integers whose squares consists of $2$ consecutive odd or even integers juxtaposed include:

$1127^2 = 01 \, 270 \, 129$
$8874^2 = 78 \, 747 \, 876$

Such integers come in pairs which add to $1$ more than a power of $10$:

$1127 + 8874 = 10 \, 001$



Proof



Historical Note

Numbers with this property were reported on by Victor Thébault, in volume $13$ of Scripta Mathematica.


Sources