ThmDex – An index of mathematical definitions, results, and conjectures.

Base-2 integer representation for minus one

Formulation 0

Let $N \in \{ 1, 2, 3, \ldots \}$ be a D5094: Positive integer.

Then \begin{equation} - 1 = - 2^{N - 1} + \sum_{n = 0}^{N - 2} 2^n \end{equation}

Formulation 1

Let $N \in \{ 1, 2, 3, \ldots \}$ be a D5094: Positive integer.

Then $\underbrace{1, \, \ldots, 1, 1}_{N \text{ times}}$ is a D6263: Integer base-2 representation for $-1$.

Formulation 2

Let $N \in \{ 1, 2, 3, \ldots \}$ be a D5094: Positive integer.

Then the following statements are equivalent

(1)	$r_{N - 1}, \, \ldots, r_1, r_0$ is a D6263: Integer base-2 representation for $-1$
(2)	\begin{equation} r_{N - 1} = \cdots = r_1 = r_0 = 1 \end{equation}