Numbers whose Squares are Consecutive Odd or Even Integers Juxtaposed

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 $10$:
 * $1127 + 8874 = 10 \, 001$