Theophilus Agama is a mathematical innovator, an inventor, an independent researcher and a problem solver. He works primarily in algebra, number theory, combinatorics, geometry and its allied areas. Supervisors: Jean-Marrie De koninck
In this short note, we study the distribution of the $\ell$ function, particularly on the primes.... more In this short note, we study the distribution of the $\ell$ function, particularly on the primes. We study various elementary properties of the $\ell$ function. We relate the $\ell$ function to prime gaps, offering a motivation for a further studies of this function.
In this paper we introduce and develop the concept of expansivity of a tuple whose entries are el... more In this paper we introduce and develop the concept of expansivity of a tuple whose entries are elements from the polynomial ring R[x]. As an inverse problem, we examine how to recover a tuple from the expanded tuple at any given phase of expansion. We convert the celebrated Sendov conjecture concerning the distribution of zeros of polynomials and their critical points into this language and prove some weak variants of this conjecture. We also apply this to the existence of solutions to differential equations. In particular, we show that a certain system of differential equation has no non-trivial solution. Motivated by the Pierce-Birkhoff conjecture, we launch an extension program for single variable expansivity theory. We study this notion under tuples of polynomials belonging to the ring R[x 1 , x 2 ,. .. , xn]. As an application we develop some class of inequalities to study the Pierce-Birkhoff conjecture
We introduce and develop the logic of existence of solution to problems. We use this theory to an... more We introduce and develop the logic of existence of solution to problems. We use this theory to answer the question of Florentin Smarandache in logic. We answer this question in the negative.
We continue with the development of the theory of problems and their solutions spaces [1] and [2]... more We continue with the development of the theory of problems and their solutions spaces [1] and [2]. We introduce and study the notion of verification and resolution time complexity of solutions and problem spaces.
Exploiting the notion of carries, we obtain improved upper bounds for the length of the shortest ... more Exploiting the notion of carries, we obtain improved upper bounds for the length of the shortest addition chains $\iota(2^n-1)$ producing $2^n-1$. Most notably, we show that if $2^n-1$ has carries of degree at most $$\kappa(2^n-1)=\frac{1}{2}(\iota(n)-\lfloor \frac{\log n}{\log 2}\rfloor+\sum \limits_{j=1}^{\lfloor \frac{\log n}{\log 2}\rfloor}\{\frac{n}{2^j}\})$$ then the inequality $$\iota(2^n-1)\leq n+1+\sum \limits_{j=1}^{\lfloor \frac{\log n}{\log 2}\rfloor}\bigg(\{\frac{n}{2^j}\}-\xi(n,j)\bigg)+\iota(n)$$ holds for all $n\in \mathbb{N}$ with $n\geq 4$, where $\iota(\cdot)$ denotes the length of the shortest addition chain producing $\cdot$, $\{\cdot\}$ denotes the fractional part of $\cdot$ and where $\xi(n,1):=\{\frac{n}{2}\}$ with $\xi(n,2)=\{\frac{1}{2}\lfloor \frac{n}{2}\rfloor\}$ and so on
A new upper bound for the Heilbronn triangle problem, 2024
Using ideas from the geometry of compression, we improve on the current upper and lower bound of ... more Using ideas from the geometry of compression, we improve on the current upper and lower bound of Heilbronn's triangle problem. In particular, by letting $\Delta(s)$ denotes the minimal area of the triangle induced by $s$ points in a unit disc, then we have the upper bound $$\Delta(s)\ll \frac{1}{s^{\frac{3}{2}-\epsilon}}$$ for small $\epsilon:=\epsilon(s)>0$.
Using the method of compression, we prove an inequality related to the Gauss circle problem. Let ... more Using the method of compression, we prove an inequality related to the Gauss circle problem. Let $\mathcal{N}_r$ denotes the number of integral points in a circle of radius $r>0$, then we have $$2r^2\bigg(1+\frac{1}{4}\sum \limits_{1\leq k\leq \lfloor \frac{\log r}{\log 2}\rfloor}\frac{1}{2^{2k-2}}\bigg)+O(\frac{r}{\log r}) \leq \mathcal{N}_r \leq 8r^{2}\bigg(1+\sum \limits_{1\leq k\leq \lfloor \frac{\log r}{\log 2}\rfloor}\frac{1}{2^{2k-2}}\bigg)+O(\frac{r}{\log r})$$ for all $r>1$. This implies that the error function $E(r)$ of the counting function $\mathcal{N}_r\ll r^{1-\epsilon}$ for any $\epsilon>0.$
In this paper we show that the number of points that can be placed in the grid n × n × • • • × n ... more In this paper we show that the number of points that can be placed in the grid n × n × • • • × n (d times) = n d for all d ∈ N with d ≥ 2 such that no three points are collinear satisfies the lower bound n d−1 2d √ d. This pretty much extends the result of the no-three-in-line problem to all dimension d ≥ 3.
In this paper, using the method of compression, we recover the lower bound for the Erdős unit dis... more In this paper, using the method of compression, we recover the lower bound for the Erdős unit distance problem and provide an alternative proof to the distinct distance conjecture. In particular, in R k for all k ≥ 2, we have # || x j − xt|| : || x j − xt|| = 1, 1 ≤ t, j ≤ n, x j , xt ∈ R k k √ k 2 n 1+o(1). We also show that # d j : d j = || xs − yt||, d j = d i , 1 ≤ s, t ≤ n k √ k 2 n 2 k −o(1). These lower bounds generalizes the lower bounds of the Erdős unit distance and the distinct distance problem to higher dimensions.
Using the method of compression we improve on the current lower bound of Heilbronn's triangle pro... more Using the method of compression we improve on the current lower bound of Heilbronn's triangle problem. In particular, by letting ∆(s) denotes the minimal area of the triangle induced by s points in a unit disc. Then we have the lower bound ∆(s) log s s √ s .
A proof of the Erdős-Turán additive base conjecture, 2024
This paper is an extension program of the notion of circle of partition developed in our first pa... more This paper is an extension program of the notion of circle of partition developed in our first paper [1]. As an application we prove the Erdős-Turán additive base conjecture.
MULTIVARIATE CIRCLE OF PARTITIONS AND THE SQUEEZE PRINCIPLE, 2023
The goal of this paper is to extend the squeeze principle to circle of partitions with at least t... more The goal of this paper is to extend the squeeze principle to circle of partitions with at least two resident points on their axes.
ON THE NOTION OF CARRIES OF NUMBERS 2^n − 1 AND SCHOLZ CONJECTURE, 2023
Applying the pothole method on the factors of numbers of the form $2^n-1$, we prove that if $2^n-... more Applying the pothole method on the factors of numbers of the form $2^n-1$, we prove that if $2^n-1$ has carries of degree at most $$\kappa(2^n-1)=\frac{1}{2(1+c)}\lfloor \frac{\log n}{\log 2}\rfloor-1$$ for $c>0$ fixed, then the inequality $$\iota(2^n-1)\leq n-1+(1+\frac{1}{1+c})\lfloor\frac{\log n}{\log 2}\rfloor$$ holds for all $n\in \mathbb{N}$ with $n\geq 4$, where $\iota(\cdot)$ denotes the length of the shortest addition chain producing $\cdot$. In general, we show that all numbers of the form $2^n-1$ with carries of degree $$\kappa(2^n-1):=(\frac{1}{1+f(n)})\lfloor \frac{\log n}{\log 2}\rfloor-1$$ with $f(n)=o(\log n)$ and $f(n)\longrightarrow \infty$ as $n\longrightarrow \infty$ for $n\geq 4$ then the inequality $$\iota(2^n-1)\leq n-1+(1+\frac{2}{1+f(n)})\lfloor\frac{\log n}{\log 2}\rfloor$$ holds.
In this short note, we study the distribution of the $\ell$ function, particularly on the primes.... more In this short note, we study the distribution of the $\ell$ function, particularly on the primes. We study various elementary properties of the $\ell$ function. We relate the $\ell$ function to prime gaps, offering a motivation for a further studies of this function.
In this paper we introduce and develop the concept of expansivity of a tuple whose entries are el... more In this paper we introduce and develop the concept of expansivity of a tuple whose entries are elements from the polynomial ring R[x]. As an inverse problem, we examine how to recover a tuple from the expanded tuple at any given phase of expansion. We convert the celebrated Sendov conjecture concerning the distribution of zeros of polynomials and their critical points into this language and prove some weak variants of this conjecture. We also apply this to the existence of solutions to differential equations. In particular, we show that a certain system of differential equation has no non-trivial solution. Motivated by the Pierce-Birkhoff conjecture, we launch an extension program for single variable expansivity theory. We study this notion under tuples of polynomials belonging to the ring R[x 1 , x 2 ,. .. , xn]. As an application we develop some class of inequalities to study the Pierce-Birkhoff conjecture
We introduce and develop the logic of existence of solution to problems. We use this theory to an... more We introduce and develop the logic of existence of solution to problems. We use this theory to answer the question of Florentin Smarandache in logic. We answer this question in the negative.
We continue with the development of the theory of problems and their solutions spaces [1] and [2]... more We continue with the development of the theory of problems and their solutions spaces [1] and [2]. We introduce and study the notion of verification and resolution time complexity of solutions and problem spaces.
Exploiting the notion of carries, we obtain improved upper bounds for the length of the shortest ... more Exploiting the notion of carries, we obtain improved upper bounds for the length of the shortest addition chains $\iota(2^n-1)$ producing $2^n-1$. Most notably, we show that if $2^n-1$ has carries of degree at most $$\kappa(2^n-1)=\frac{1}{2}(\iota(n)-\lfloor \frac{\log n}{\log 2}\rfloor+\sum \limits_{j=1}^{\lfloor \frac{\log n}{\log 2}\rfloor}\{\frac{n}{2^j}\})$$ then the inequality $$\iota(2^n-1)\leq n+1+\sum \limits_{j=1}^{\lfloor \frac{\log n}{\log 2}\rfloor}\bigg(\{\frac{n}{2^j}\}-\xi(n,j)\bigg)+\iota(n)$$ holds for all $n\in \mathbb{N}$ with $n\geq 4$, where $\iota(\cdot)$ denotes the length of the shortest addition chain producing $\cdot$, $\{\cdot\}$ denotes the fractional part of $\cdot$ and where $\xi(n,1):=\{\frac{n}{2}\}$ with $\xi(n,2)=\{\frac{1}{2}\lfloor \frac{n}{2}\rfloor\}$ and so on
A new upper bound for the Heilbronn triangle problem, 2024
Using ideas from the geometry of compression, we improve on the current upper and lower bound of ... more Using ideas from the geometry of compression, we improve on the current upper and lower bound of Heilbronn's triangle problem. In particular, by letting $\Delta(s)$ denotes the minimal area of the triangle induced by $s$ points in a unit disc, then we have the upper bound $$\Delta(s)\ll \frac{1}{s^{\frac{3}{2}-\epsilon}}$$ for small $\epsilon:=\epsilon(s)>0$.
Using the method of compression, we prove an inequality related to the Gauss circle problem. Let ... more Using the method of compression, we prove an inequality related to the Gauss circle problem. Let $\mathcal{N}_r$ denotes the number of integral points in a circle of radius $r>0$, then we have $$2r^2\bigg(1+\frac{1}{4}\sum \limits_{1\leq k\leq \lfloor \frac{\log r}{\log 2}\rfloor}\frac{1}{2^{2k-2}}\bigg)+O(\frac{r}{\log r}) \leq \mathcal{N}_r \leq 8r^{2}\bigg(1+\sum \limits_{1\leq k\leq \lfloor \frac{\log r}{\log 2}\rfloor}\frac{1}{2^{2k-2}}\bigg)+O(\frac{r}{\log r})$$ for all $r>1$. This implies that the error function $E(r)$ of the counting function $\mathcal{N}_r\ll r^{1-\epsilon}$ for any $\epsilon>0.$
In this paper we show that the number of points that can be placed in the grid n × n × • • • × n ... more In this paper we show that the number of points that can be placed in the grid n × n × • • • × n (d times) = n d for all d ∈ N with d ≥ 2 such that no three points are collinear satisfies the lower bound n d−1 2d √ d. This pretty much extends the result of the no-three-in-line problem to all dimension d ≥ 3.
In this paper, using the method of compression, we recover the lower bound for the Erdős unit dis... more In this paper, using the method of compression, we recover the lower bound for the Erdős unit distance problem and provide an alternative proof to the distinct distance conjecture. In particular, in R k for all k ≥ 2, we have # || x j − xt|| : || x j − xt|| = 1, 1 ≤ t, j ≤ n, x j , xt ∈ R k k √ k 2 n 1+o(1). We also show that # d j : d j = || xs − yt||, d j = d i , 1 ≤ s, t ≤ n k √ k 2 n 2 k −o(1). These lower bounds generalizes the lower bounds of the Erdős unit distance and the distinct distance problem to higher dimensions.
Using the method of compression we improve on the current lower bound of Heilbronn's triangle pro... more Using the method of compression we improve on the current lower bound of Heilbronn's triangle problem. In particular, by letting ∆(s) denotes the minimal area of the triangle induced by s points in a unit disc. Then we have the lower bound ∆(s) log s s √ s .
A proof of the Erdős-Turán additive base conjecture, 2024
This paper is an extension program of the notion of circle of partition developed in our first pa... more This paper is an extension program of the notion of circle of partition developed in our first paper [1]. As an application we prove the Erdős-Turán additive base conjecture.
MULTIVARIATE CIRCLE OF PARTITIONS AND THE SQUEEZE PRINCIPLE, 2023
The goal of this paper is to extend the squeeze principle to circle of partitions with at least t... more The goal of this paper is to extend the squeeze principle to circle of partitions with at least two resident points on their axes.
ON THE NOTION OF CARRIES OF NUMBERS 2^n − 1 AND SCHOLZ CONJECTURE, 2023
Applying the pothole method on the factors of numbers of the form $2^n-1$, we prove that if $2^n-... more Applying the pothole method on the factors of numbers of the form $2^n-1$, we prove that if $2^n-1$ has carries of degree at most $$\kappa(2^n-1)=\frac{1}{2(1+c)}\lfloor \frac{\log n}{\log 2}\rfloor-1$$ for $c>0$ fixed, then the inequality $$\iota(2^n-1)\leq n-1+(1+\frac{1}{1+c})\lfloor\frac{\log n}{\log 2}\rfloor$$ holds for all $n\in \mathbb{N}$ with $n\geq 4$, where $\iota(\cdot)$ denotes the length of the shortest addition chain producing $\cdot$. In general, we show that all numbers of the form $2^n-1$ with carries of degree $$\kappa(2^n-1):=(\frac{1}{1+f(n)})\lfloor \frac{\log n}{\log 2}\rfloor-1$$ with $f(n)=o(\log n)$ and $f(n)\longrightarrow \infty$ as $n\longrightarrow \infty$ for $n\geq 4$ then the inequality $$\iota(2^n-1)\leq n-1+(1+\frac{2}{1+f(n)})\lfloor\frac{\log n}{\log 2}\rfloor$$ holds.
We apply the notion of the \textbf{olloid} to show that the family of Erd\H{o}s-Straus type equat... more We apply the notion of the \textbf{olloid} to show that the family of Erd\H{o}s-Straus type equation $$\frac{4^{2^l}}{n^{2^l}}=\frac{1}{x^{2^l}}+\frac{1}{y^{2^l}}+\frac{1}{z^{2^l}}$$ has solutions for all $l\geq 1$ provided the equation $$\frac{4}{n}=\frac{1}{x}+\frac{1}{y}+\frac{1}{z}$$ has solution for a fixed $n>4$.
THE ASYMPTOTIC SQUEEZE PRINCIPLE AND THE BINARY GOLDBACH CONJECTURE, 2023
In this paper, we prove the special squeeze principle for all sufficiently large n ∈ 2N. This pro... more In this paper, we prove the special squeeze principle for all sufficiently large n ∈ 2N. This provides an alternative proof for the asymptotic version of the binary Goldbach conjecture in [3].
ON THE GEOMETRY OF AXES OF COMPLEX CIRCLES OF PARTITION -PART 1, 2023
In this paper we continue the development of the circles of partition by introducing a certain ge... more In this paper we continue the development of the circles of partition by introducing a certain geometry of the axes of complex circles of partition. We use this geometry to verify the condition in the squeeze principle in special cases with regards to the orientation of the axes of complex circles of partition.
ON THE TOPOLOGY OF PROBLEMS AND THEIR SOLUTIONS, 2023
In this paper, we study the topology of problems and their solution spaces developed introduced i... more In this paper, we study the topology of problems and their solution spaces developed introduced in our first paper [1]. We introduce and study the notion of separability and quotient problem and solution spaces. This notions will form a basic underpinning for further studies on this topic.
ON SOLUTIONS TO ERDŐS-STRAUS EQUATION OVER CERTAIN INTEGER POWERS, 2023
We apply the notion of the \textbf{olloid} to show that the Erd\H{o}s-Straus equation $$\frac{4}{... more We apply the notion of the \textbf{olloid} to show that the Erd\H{o}s-Straus equation $$\frac{4}{n^{2^l}}=\frac{1}{x}+\frac{1}{y}+\frac{1}{z}$$ has solutions for all $l\geq 1$ provided the equation $$\frac{4}{n}=\frac{1}{x}+\frac{1}{y}+\frac{1}{z}$$ has solution for a fixed $n>4$.
IMPROVED BOUND FOR THE NUMBER OF INTEGRAL POINTS IN A CIRCLE OF RADIUS r > 1, 2022
Using the method of compression, we prove an inequality related to the Gauss circle problem. Let ... more Using the method of compression, we prove an inequality related to the Gauss circle problem. Let $\mathcal{N}_r$ denotes the number of integral points in a circle of radius $r>0$, then we have $$2r^2\bigg(1+\frac{1}{4}\sum \limits_{1\leq k\leq \lfloor \frac{\log r}{\log 2}\rfloor}\frac{1}{2^{2k-2}}\bigg)+O(\frac{r}{\log r}) \leq \mathcal{N}_r \leq 8r^{2}\bigg(1+\sum \limits_{1\leq k\leq \lfloor \frac{\log r}{\log 2}\rfloor}\frac{1}{2^{2k-2}}\bigg)+O(\frac{r}{\log r})$$ for all $r>1$.
A PROOF OF THE SCHOLZ CONJECTURE ON ADDITION CHAINS, 2022
Applying the pothole method on the factors of numbers of the form 2 n − 1, we prove the inequalit... more Applying the pothole method on the factors of numbers of the form 2 n − 1, we prove the inequality ι(2 n − 1) ≤ n − 1 + ι(n) where ι(n) denotes the length of the shortest addition chain producing n.
A REFINED POTHOLE METHOD AND THE SCHOLZ CONJECTURE ON ADDITION CHAINS, 2022
Applying the pothole method on the factors of numbers of the form $2^n-1$, we prove the inequalit... more Applying the pothole method on the factors of numbers of the form $2^n-1$, we prove the inequality $$\iota(2^n-1)\leq \frac{3}{2}n-\left \lfloor \frac{n-2}{2^{\lfloor \frac{\log n}{\log 2}-1\rfloor+1}}\right \rfloor-\lfloor \frac{\log n}{\log 2}-1\rfloor +\frac{1}{4}(1-(-1)^n)+\iota(n)$$ where $\lfloor \cdot \rfloor$ denotes the floor function and $\iota(n)$ the shortest addition chain producing $n$.
ON THE THEORY OF PROBLEMS AND THEIR SOLUTION SPACES, 2022
We introduce and develop the logic of existence of solution to problems. We use this theory to an... more We introduce and develop the logic of existence of solution to problems. We use this theory to answer the question of Florentin Smarandache in logic. We answer this question in the negative.
ON A CONJECTURE OF ERDŐS ON ADDITIVE BASIS OF LARGE ORDERS, 2022
Using the methods of multivariate circles of partition, we prove that for any additive base $\mat... more Using the methods of multivariate circles of partition, we prove that for any additive base $\mathbb{A}$ of order $h\geq 2$ the upper bound $$\# \left \{(x_1,x_2,\ldots,x_h)\in \mathbb{A}^h~|~\sum \limits_{i=1}^{h}x_i=k\right \}\ll_{h}\log k$$ holds for sufficiently large values of $k$ provided the counting function $$\# \left \{(x_1,x_2,\ldots,x_h)\in \mathbb{A}^h~|~\sum \limits_{i=1}^{h}x_i=k\right \}$$ is an increasing function for all $k$ sufficiently large.
ON THE DISTRIBUTION OF PERFECT NUMBERS AND RELATED SEQUENCES VIA THE NOTION OF THE DISC, 2022
In this paper we investigate some properties of perfect numbers and associated sequences using th... more In this paper we investigate some properties of perfect numbers and associated sequences using the notion of the disc induced by the sum-ofthe-divisor function σ. We reveal an important relationship between perfect numbers and abundant numbers.
COMPLEX CIRCLES OF PARTITION AND THE ASYMPTOTIC LEMOINE CONJECTURE, 2022
Using the methods of the complex circles of partition (cCoPs), we study interior and exterior poi... more Using the methods of the complex circles of partition (cCoPs), we study interior and exterior points of such structures in the complex plane. With simitarities to quotient groups inside of the group theory we define quotient cCoPs. With it we can prove an asymptotic version of the Lemoine Conjecture.
COMPLEX CIRCLES OF PARTITION AND THE ASYMPTOTIC BINARY GOLDBACH CONJECTURE, 2022
In this work, we continue the complex circle of partition development that was started in our fou... more In this work, we continue the complex circle of partition development that was started in our foundational study [3]. With regard to a cCoP and its embedding circle, we define interior and exterior points. On this foundation, we expand the concept of point density, established in [2], to include complex circles of partition. We propose the idea of a quotient complex circle of partition and investigate some of its features in analogy to the quotient group in group theory. With this notion we can prove an asymptotic version of the Binary Goldbach Conjecture.
ON A SET AVOIDING SOLUTIONS OF THE ERDŐS-STRAUS EQUATION, 2022
We apply the notion of the olloid to show that a certain set contains no solution of the Erdős-St... more We apply the notion of the olloid to show that a certain set contains no solution of the Erdős-Straus equation.
A MULTIVARIATE ANALOGUE OF JENSEN'S INEQUALITY VIA THE LOCAL PRODUCT SPACE, 2022
In this note we prove a multivariate analogue of Jensen's inequality via the notion of the local ... more In this note we prove a multivariate analogue of Jensen's inequality via the notion of the local product and associated space.
ON THE GENERAL ERDŐS-MOSER EQUATION VIA THE NOTION OF OLLOIDS, 2022
We introduce and develop the notion of the olloid. We apply this notion to study a variant and a ... more We introduce and develop the notion of the olloid. We apply this notion to study a variant and a generalized version of the Erdős-Moser equation under some special local condition.
THE NOTION OF OLLOIDS AND THE ERDŐS-MOSER EQUATION, 2022
We introduce and develop the notion of the olloid. We apply this notion to study the Erdős-Moser ... more We introduce and develop the notion of the olloid. We apply this notion to study the Erdős-Moser equation.
In this paper we show that the shortest length $\iota(n)$ of addition chains producing numbers of... more In this paper we show that the shortest length $\iota(n)$ of addition chains producing numbers of the form $2^n-1$ satisfies the lower bound $$\iota(2^n-1)\geq n+\lfloor \frac{\log (n-1)}{\log 2}\rfloor$$ where $\lfloor \cdot \rfloor$ denotes the floor function.
AN IMPROVED BOUND FOR LENGTH OF ADDITION CHAINS PRODUCING 2 n − 1, 2022
In this paper we prove that there exists an addition chain producing $2^n-1$ of length $\delta(2^... more In this paper we prove that there exists an addition chain producing $2^n-1$ of length $\delta(2^n-1)$ satisfying the inequality \begin{align} \delta(2^n-1)\leq 2n-1-\left \lfloor \frac{n-1}{2^{\lfloor \frac{\log n}{\log 2}\rfloor}}\right \rfloor-\lfloor \frac{\log n}{\log 2}\rfloor +\iota(n)\nonumber \end{align}where $\lfloor \cdot \rfloor$ denotes the floor function and $\iota(n)$ the shortest addition chain producing $n$.
In this paper we prove that there exists an addition chain producing $2^n-1$ of length $\delta(2^... more In this paper we prove that there exists an addition chain producing $2^n-1$ of length $\delta(2^n-1)$ satisfying the inequality \begin{align} \delta(2^n-1)\leq 2n-1-2\left \lfloor \frac{n-1}{2^{\lfloor \frac{\log n}{\log 2}\rfloor}}\right \rfloor+\lfloor \frac{\log n}{\log 2}\rfloor\nonumber \end{align}where $\lfloor \cdot \rfloor$ denotes the floor function.
Uploads
Papers by Theophilus Agama
where $\lfloor \cdot \rfloor$ denotes the floor function.
\begin{align}
\delta(2^n-1)\leq 2n-1-\left \lfloor \frac{n-1}{2^{\lfloor \frac{\log n}{\log 2}\rfloor}}\right \rfloor-\lfloor \frac{\log n}{\log 2}\rfloor +\iota(n)\nonumber
\end{align}where $\lfloor \cdot \rfloor$ denotes the floor function and $\iota(n)$ the shortest addition chain producing $n$.
\begin{align}
\delta(2^n-1)\leq 2n-1-2\left \lfloor \frac{n-1}{2^{\lfloor \frac{\log n}{\log 2}\rfloor}}\right \rfloor+\lfloor \frac{\log n}{\log 2}\rfloor\nonumber
\end{align}where $\lfloor \cdot \rfloor$ denotes the floor function.