Mohammad K. Azarian | Searcharxiv

arXiv2026

On the History of the Square and Multiply Algorithm

The square-and-multiply algorithm, also known as binary exponentiation or repeated squaring, is a standard method for fast exponentiation in modern computation. Its historical origins, however, remain uncertain. This paper examines the emergence and progressive formalization of the method through selected primary sources. Particular attention is given to Jamshid al-Kashi's fifteenth-century Miftah al-Hisab, where the procedure is presented explicitly as a general computational method and claimed by al-Kashi as his own innovation. Earlier instances of successive squaring are identified in the works of al-Uqlidisi and al-Biruni, although in these cases the technique appears in particular calculations rather than as a fully articulated general rule. The earliest known antecedent is found in Pingala's prosodic studies in ancient India (c. 200 BCE), which seem to presuppose the conceptual basis of the method in their use of binary representation. As part of the historical development, Legendre's 1798 worked example is one of the earliest documented European use of the algorithm which appears as a subordinate step within a specific number-theoretic computation. The evidence suggests not a single continuous line of transmission, but the repeated independent reappearance of related procedures in distinct contexts. By the twentieth century, square-and-multiply became a special case within the broader theory of addition chains. By exploring this intellectual progression, this paper sheds some light on the historical background of an algorithm that is prominent in modern computation.

math.HO↗

arXiv2026

Gap-Sums via Quasi-Arithmetic Means with Applications to Fibonacci and Lucas Sequences

We develop a unified framework for studying the integers missing between consecutive terms of an increasing integer sequence, extending Barry's arithmetic gap-sum to geometric and harmonic analogues via the theory of quasi-arithmetic means. All three gap-sums admit a common interpretation: each equals the gap size multiplied by the appropriate mean of the missing integers. Building on this, we prove a general sparse summation theorem expressing the sum of a strictly monotonic function over a sparse integer sequence as the full range sum corrected by the gap-sums of the missing portions. Specializing on the three Pythagorean means recovers a classical formula of al-Kāsh\=ı from the fifteenth century in the arithmetic case, and yields explicit formulas in the geometric and harmonic cases. As a concrete application of the geometric case, we derive a product identity involving the Fuss--Catalan numbers. Applying the harmonic case to the Fibonacci and Lucas sequences, we establish that the harmonic gap-sum converges to $\ln(α)$ exponentially, where $α$ is the golden ratio, and derive explicit two-term asymptotic expansions for the tails of the reciprocal Fibonacci and Lucas series with closed-form coefficients, and establish the asymptotic formula $H_{u_n} \sim n\ln(α)$ for both $u_n = F_n$ and $u_n = L_n$, with explicit $O(1)$ error terms that differ due to their distinct initial conditions. As a further consequence, by comparing the gap-sum expansions with the classical Hardy--Wright expansion of harmonic numbers, we derive exact series identities expressing Euler's constant $γ$ in terms of harmonic numbers at Fibonacci and Lucas indices, and obtain a new identity relating the reciprocal Fibonacci constant $ψ$ and the reciprocal Lucas constant $ψ_L$.

math.NT↗