Papers by Theophilus Agama
THE SCHOLZ CONJECTURE IS TRUE FOR 2^n -1 FOR ALMOST ALL n, 2026
THEOPHILUS AGAMA A. An addition chain of length h that leads to a number n is a sequence of posit... more THEOPHILUS AGAMA A. An addition chain of length h that leads to a number n is a sequence of positive integers s_0 = 1, s_1 = 2,. .. , s_h = n such that s_i = s_j + s_k (i > j ≥ k) for each 1 ≤ i ≤ h. A Brauer addition chain is the one where j = i-1 for each 1 ≤ i ≤ h. Let l(•) and l* (•) denote the minimal length of an addition chain and the Brauer addition chain, respectively, that leads to an integer •. Applying probabilistic methods to the iterated factor method, we show that l(2 n-1) ≤ n-1 + l(n) for almost all positive integers n as n-→ ∞.
PROBLEM THEORY AND APPLICATION TO THE P VS NP PROBLEM, 2026
This paper introduces problem theory, a new framework for studying problem and solution spaces th... more This paper introduces problem theory, a new framework for studying problem and solution spaces through the lens of point-set topology and abstract algebra. We define solution spaces as topological constructs induced by the assignment of solutions to problems and establish their fundamental properties. Key results include the identification of compactness and continuity conditions in solution spaces and their algebraic interpretations within module-theoretic settings. This theory bridges abstract algebra and topology, providing new insights into the interplay between algebraic structures and topological spaces. Potential applications and directions for future research are discussed.
ON THE SCHOLZ-BRAUER PROBLEM, 2026
We study structural conditions on addition chains that guarantee the classical Scholz-Brauer ineq... more We study structural conditions on addition chains that guarantee the classical Scholz-Brauer inequality (2 n-1) ≤ n-1 + (n), where (m) denotes the minimal length of an addition chain for (m). After recalling the standard decomposition s i = s σ(i) + s τ (i) of a chain, we introduce two combinatorial closure properties for exponent chains: a closed chain (gaps and nonzero forward differences along the σ-track occur among chain terms) and the weaker absolutely closed chain (the same condition up to absolute value). An integer n is called complete (resp. absolutely complete) when it admits an optimal closed (resp. absolutely closed) chain. Our main results are constructive and threefold. First, every complete number satisfies the Scholz-Brauer inequality; the proof lifts an optimal closed exponent chain to a sequence of Mersenne seeds 2 si-1 and uses the built-in gaps of the chain to schedule canonical doublings, producing an explicit addition chain for 2 n-1 of length at most n-1 + (n). Second, the same lifting argument extends to absolutely complete numbers, yielding an identical bound. Third, we present a broadly applicable structural criterion existence of an optimal chain with nondecreasing σ-track, no two consecutive non-Brauer steps, and no three consecutive equal σ-track values which also implies the Scholz-Brauer inequality; this condition subsumes many natural non-Brauer optimal chains. The paper unifies and generalizes classical Brauer/Hansen doubling constructions, provides explicit examples and constructions, and suggests effective checks to detect integers for which the Scholz-Brauer bound holds.
BOUNDS FOR MINIMAL LENGTH OF FIXED DEGREE CHAINS, 2026
We study addition chains of fixed degree $d\geq 2$, i.e. sequences in which each term is the sum ... more We study addition chains of fixed degree $d\geq 2$, i.e. sequences in which each term is the sum of at most $d$ earlier terms, and we establish explicit bounds on the minimal length $\ell^d(n)$ of such chains leading to an integer $n$. For an upper bound, we give constructive Brauer-type bounds: for $d^m\leq n<d^{m+1}$, we prove
$$
\ell^d(n)\leq \min_{1\leq r\leq v\leq m}\bigg[\big(1+\tfrac{1}{r}\big)\frac{\log n}{\log d}+d^v-2\bigg],
$$
and a refined construction yielding
$$
\ell^d(n)\leq \min_{1\leq r\leq m/(d-1)}\bigg[\bigg(1+\frac{1}{(d-1)r}\bigg)\frac{\log n}{\log d}+(d-1)(d^r-1)\bigg].
$$
For a lower bound, we prove a Sch\"onhage -type bound: writing $\nu_d(n)$ for the number of nonzero base-$d$ digits of $n$ and $\gamma_d=\frac{d-1+\sqrt{(d-1)^2+4}}{2}$, any fixed-degree chain satisfies
$$
\ell^d(n)\geq \frac{\log n}{\log d}+\bigg(1-\frac{\log \gamma_d}{\log d}\bigg)\frac{\log\nu_d(n)}{\log d}.
$$
The upper bounds are obtained by explicit chain constructions, while the lower bound follows from combinatorial and digit-counting arguments using a partition into step framework adapted to the fixed degree $d\geq 2$ framework.
PRODUCT DISTRIBUTION IN A FIXED DEGREE CHAIN, 2026
We study multiplicative structures arising from \emph{fixed-degree} addition chains: increasing s... more We study multiplicative structures arising from \emph{fixed-degree} addition chains: increasing sequences
$$
s_0=1<s_1<\cdots<s_h
$$
in which each term is the sum of at most $d\geq 2$ earlier terms (repetition allowed). Writing $\mathcal{P}(L)$ for the set of integers that can be expressed as a product of distinct terms drawn from a (finite or infinite) fixed-degree chain $L=\{T_m\}$, we develop a structural partition of chain steps into four classes $\mathcal{A}_d,\mathcal{B}_d,\mathcal{C}_d,\mathcal{D}_d$ (dilate / large / medium / small) and introduce the notion of a \emph{run} of a given step type. By deriving precise two-sided log-sum (product) estimates along pure and mixed-type runs and translating these into elementary counting bounds (for example, run-level bounds of the shape $\ll(\log x)^{1/2}$ in the logarithmic scale), we obtain a global sparsity result: under the mild structural hypothesis that the number of runs of types $\mathcal{D}_d$ and $\mathcal{A}_d\cup\mathcal{D}_d$ on initial segments of the chain is $O_d(1)$, the product-set $\mathcal{P}(L)$ has zero asymptotic density. In particular, for any such infinite chain $L$ one has
$$
\lim_{x\longrightarrow \infty}\frac{\#\{v\leq x~:~v\in\mathcal{P}(L)\}}{x}=0.
$$
The proof combines combinatorial run-analysis, elementary exponential growth estimates along runs, and an assembly of run-level contributions into explicit global counting bounds; quantitative run-level inequalities (involving the clearly defined parameters $\lambda_d$ and a slowly varying $\delta(m)=1/\log m$) are recorded and may be of independent interest for further study of fixed-degree chains.
MINIMAL LENGTH OF FIXED DEGREE S-RICH CHAINS, 2026
We introduce and study $\mathbb{S}$-rich fixed-degree addition chains of a fixed index $n>1$: inc... more We introduce and study $\mathbb{S}$-rich fixed-degree addition chains of a fixed index $n>1$: increasing sequences of positive integers beginning at $1$ in which every term is the sum of at most $d$ earlier terms and which contain all elements of the set $\mathbb{S}\cap [1,n]$ as targets. Our main contribution is a simple constructive upper bound for the minimal length $\ell_n^d(\mathbb{S})$ of such a chain. Writing the ordered elements of $\mathbb{S} \cap [1,n]$ as $z_1\leq z_2\leq \cdots$ and letting $\nu_d(\cdot)$ denote the sum of base-$d$ digits, we show
$$
\ell_{n}^d(\mathbb{S})\leq \left \lfloor\frac{\log n}{\log d}\right \rfloor+\left \lceil \frac{\nu_d(z_1)}{d-1}\right \rceil+\sum \limits_{i=2}^{|\mathbb{S}_n|}\left \lceil \frac{\nu_d(z_i-z_{i-1})}{d-1} \right \rceil.
$$
a bound that generalizes the standard base-expansion construction for single targets and mirrors the digit-sum phenomenon appearing in known lower bounds. The proof is constructive and yields a straightforward algorithm for chain construction; we record corollaries for prime-rich and square-free sets, give illustrative examples, and discuss tightness, and natural open problems about optimal multi-target precomputation.
ON THE FIXED DEGREE SCHOLZ-BRAUER PROBLEM, 2026
Denote the minimal length of a fixed degree d ≥ 2 addition chain that leads to n by d (n). We int... more Denote the minimal length of a fixed degree d ≥ 2 addition chain that leads to n by d (n). We introduce the concept of a strong Brauer number of rank d ≥ 2 and show that all numbers belonging to this class satisfy the inequality
$$
\ell^d (d^n-1) ≤ n-1 + \ell^d (n).
$$
This extends the concept of a Brauer number in standard addition chain theory to the fixed degree d ≥ 2 framework.
ON THE GENERALIZED RUN LENGTH-DENSITY THEOREM, 2026
We generalize the run length-density theorem on addition chains, an assertion that infinite seque... more We generalize the run length-density theorem on addition chains, an assertion that infinite sequences whose finite truncations constitute an addition chain and with a certain relatively slowly growing consecutive steps having short stretches must have a zero logarithmic density.
COUNTING FIXED DEGREE d ≥ 2 ADDITION CHAINS, 2026
We denote the length of an addition chain with fixed degree d ≥ 2 leading to n by l d (n). We stu... more We denote the length of an addition chain with fixed degree d ≥ 2 leading to n by l d (n). We study the counting function F d (m, r) := #{n ∈ [d m , d m+1) : l d (n) ≤ m + r} establishing upper and lower bounds, which generalizes previous classical investigations of De Koninck, Doyon, and Verreault.
ON THE GENERALIZED RUN LENGTH-GAP THEOREM, 2026
We extend the run length-gap theorem previously proved to the context of infinite sequences where... more We extend the run length-gap theorem previously proved to the context of infinite sequences where each term is built using at most d previous terms for a fixed d ≥ 2.
GENERALIZED CRUDE BRAUER INEQUALITY ON ADDITION CHAINS, 2025
We extend the inequality due to Alfred Brauer on standard addition chains to a sequence of additi... more We extend the inequality due to Alfred Brauer on standard addition chains to a sequence of additions leading to a finite number where at most at most d ≥ 2 previous terms can be added to generate each term in the sequence.
SOME PROBLEMS AND CONJECTURES ON ADDITION CHAINS, 2025
In our studies, we compile a list of problems and conjectures on the subject of addition chains. ... more In our studies, we compile a list of problems and conjectures on the subject of addition chains. These may not be heavyweight problems, but serve as a natural research direction from our work.
A LOWER BOUND FOR THE ENTROPY OF AN ADDITION CHAIN, 2025
We prove a non trivial lower bound for the entropy of an addition chain leading to a given target.
A PROGRESS ON THE SCHOLZ CONJECTURE ON ADDITION CHAINS, 2025
We prove the Scholz conjecture for all integers for which there exists an optimal addition chain ... more We prove the Scholz conjecture for all integers for which there exists an optimal addition chain with nondecreasing σ track such that no three consecutive terms in the chain have the same σ track term and that no two consecutive steps are non-Brauer steps. This work advances our previous investigation of the conjecture, where it was established for a certain class of addition chains.
SOME STRUCTURAL INEQUALITIES OF STEPS IN A CHAIN, 2025
We exploit the anatomy of Brauer and non-Brauer steps in an addition chain to prove some structur... more We exploit the anatomy of Brauer and non-Brauer steps in an addition chain to prove some structural inequalities of steps in a chain.
ON THE EXISTENCE OF CHAIN TERMS IN SMALL INTERVALS, 2025
In this paper, we exploit the anatomy of the steps introduced and developed to study the existenc... more In this paper, we exploit the anatomy of the steps introduced and developed to study the existence of terms in an addition chain in specific intervals determined by terms with σ track indices.
COUNTING INTEGERS AS PRODUCT OF TERMS IN A CHAIN, 2025
We count the number of integers t ≤ n that can be written as the product of terms in an addition ... more We count the number of integers t ≤ n that can be written as the product of terms in an addition chain leading to a specified target. Consequently, we deduce that the natural density of the set of integers with this property is zero.
ON THE ANATOMY OF BRAUER AND NON-BRAUER STEPS IN A CHAIN, 2025
We develop the anatomy of steps in an addition chain.
ON PROBLEMS IN ADDITION CHAINS WITH EXACT FIXED DEGREE, 2025
We gather and organize a rich collection of nontrivial open problems concerning the cumulative ex... more We gather and organize a rich collection of nontrivial open problems concerning the cumulative exact fixed-degree model for constructing addition chains that lead to targets of the form 2 n-1. The model generalizes classical addition chains by allowing steps that add exactly a specified number of previous terms (degrees), subject to an additive constraint on chosen degrees. The problems span structural, extremal, asymptotic, algorithmic, probabilistic, and computational complexity directions. Each problem is clearly stated and supplemented with comments, suggested first steps, and suggestions for experimentation. This document is intended as a compact research agenda and a reference for further work on this model.
THE ENTROPY METHOD IN ADDITION CHAINS, 2025
We introduce the entropy method as a generalization of the energy method introduced and studied i... more We introduce the entropy method as a generalization of the energy method introduced and studied in our previous investigations. The key feature of this method allows us to bound the optimal length of an addition chain for numbers that have all 1 s in their binary representation by two components, namely the entropy and the optimal length of the chain.
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Papers by Theophilus Agama
$$
\ell^d(n)\leq \min_{1\leq r\leq v\leq m}\bigg[\big(1+\tfrac{1}{r}\big)\frac{\log n}{\log d}+d^v-2\bigg],
$$
and a refined construction yielding
$$
\ell^d(n)\leq \min_{1\leq r\leq m/(d-1)}\bigg[\bigg(1+\frac{1}{(d-1)r}\bigg)\frac{\log n}{\log d}+(d-1)(d^r-1)\bigg].
$$
For a lower bound, we prove a Sch\"onhage -type bound: writing $\nu_d(n)$ for the number of nonzero base-$d$ digits of $n$ and $\gamma_d=\frac{d-1+\sqrt{(d-1)^2+4}}{2}$, any fixed-degree chain satisfies
$$
\ell^d(n)\geq \frac{\log n}{\log d}+\bigg(1-\frac{\log \gamma_d}{\log d}\bigg)\frac{\log\nu_d(n)}{\log d}.
$$
The upper bounds are obtained by explicit chain constructions, while the lower bound follows from combinatorial and digit-counting arguments using a partition into step framework adapted to the fixed degree $d\geq 2$ framework.
$$
s_0=1<s_1<\cdots<s_h
$$
in which each term is the sum of at most $d\geq 2$ earlier terms (repetition allowed). Writing $\mathcal{P}(L)$ for the set of integers that can be expressed as a product of distinct terms drawn from a (finite or infinite) fixed-degree chain $L=\{T_m\}$, we develop a structural partition of chain steps into four classes $\mathcal{A}_d,\mathcal{B}_d,\mathcal{C}_d,\mathcal{D}_d$ (dilate / large / medium / small) and introduce the notion of a \emph{run} of a given step type. By deriving precise two-sided log-sum (product) estimates along pure and mixed-type runs and translating these into elementary counting bounds (for example, run-level bounds of the shape $\ll(\log x)^{1/2}$ in the logarithmic scale), we obtain a global sparsity result: under the mild structural hypothesis that the number of runs of types $\mathcal{D}_d$ and $\mathcal{A}_d\cup\mathcal{D}_d$ on initial segments of the chain is $O_d(1)$, the product-set $\mathcal{P}(L)$ has zero asymptotic density. In particular, for any such infinite chain $L$ one has
$$
\lim_{x\longrightarrow \infty}\frac{\#\{v\leq x~:~v\in\mathcal{P}(L)\}}{x}=0.
$$
The proof combines combinatorial run-analysis, elementary exponential growth estimates along runs, and an assembly of run-level contributions into explicit global counting bounds; quantitative run-level inequalities (involving the clearly defined parameters $\lambda_d$ and a slowly varying $\delta(m)=1/\log m$) are recorded and may be of independent interest for further study of fixed-degree chains.
$$
\ell_{n}^d(\mathbb{S})\leq \left \lfloor\frac{\log n}{\log d}\right \rfloor+\left \lceil \frac{\nu_d(z_1)}{d-1}\right \rceil+\sum \limits_{i=2}^{|\mathbb{S}_n|}\left \lceil \frac{\nu_d(z_i-z_{i-1})}{d-1} \right \rceil.
$$
a bound that generalizes the standard base-expansion construction for single targets and mirrors the digit-sum phenomenon appearing in known lower bounds. The proof is constructive and yields a straightforward algorithm for chain construction; we record corollaries for prime-rich and square-free sets, give illustrative examples, and discuss tightness, and natural open problems about optimal multi-target precomputation.
$$
\ell^d (d^n-1) ≤ n-1 + \ell^d (n).
$$
This extends the concept of a Brauer number in standard addition chain theory to the fixed degree d ≥ 2 framework.