<s>[INST] Does the fact that $x^2=(x-1)(x+1)+1$ have a name? Just curious about this pattern
$$x^2 = (x-1)(x+1) +1$$
So:
$$\begin{align}
1^2 &= \phantom{1}0\cdot\phantom{1}2+1 = 1 \\
    2^2 &= \phantom{1}1\cdot\phantom{1}3+1 = 4 \\
    3^2 &= \phantom{1}2\cdot\phantom{1}4+1 = 9 \\
    4^2 &= \phantom{1}3\cdot\phantom{1}5+1 = 16 \\
    9^2 &= \phantom{1}8\cdot10+1 = 81 \\
    15^2 &= 14\cdot16+1 = 225
\end{align}$$
    and so on.
Then, to know any number raised to the power of $2$, you can multiply the previous number $(x-1)$ by the next one $(x+1)$, and add $1$.
So, to know the squared root of a number like $64$, you have to substract $1$ ($63$) and look for two numbers multiplied are $63$ and subtracted are $2$:
$$x \cdot y = 63 \qquad x - y = 2$$
Solving the equation you get $9$ and $7$ (or $-7$ and $-9$). The number between these is the square root ($8$).
I don't know if this apply for power of $3$.

Does this fact/theorem/relation has a name or something?

 [/INST] A similar formula for $x^3$ is $$x^3=(x-1)(x^2+x+1)+1$$
You can extend this to higher powers
$$x^n= (x-1)(x^{n-1}+x^{n-2}+...+1)+1$$
 Which is the basis for the sum of  geometric series formula.
$$1+x+x^2+...+x^{n} = \frac {x^{n+1}-1}{x-1}$$
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