arXiv subjects

Joseph M. Shunia

Publications and source records attributed to Joseph M. Shunia.

11 recordsLinked to original sources

arXiv2026

Cyclotomic Prime Extractors

We develop explicit prime-recovery formulas from the values $Φ_n(2)$ of cyclotomic polynomials. Binary divisibility patterns detect repeated prime factors and identify the least prime divisor of a squarefree index, while small corrections to $\log_2Φ_n(2)$ allow successive recovery of the distinct prime factors. Specializing the index gives identities for prime products and the least prime above a given integer. The same mechanism extends from finite factorizations to an infinite prime sequence: a normalized limit of cyclotomic values along the odd primorials defines a real constant $Ω=0.25061403238015047218\ldots$, from which every odd prime can be recovered by a recursive rounding rule.

math.NT↗

arXiv2026

Prime-Interval Algebras

Starting from a positive integer $n$ and no a priori information about the primes above it, we construct a polynomial quotient ring that recovers exactly the primes in $(n,2n]$ from a single modular exponentiation. The primes occur simultaneously as the nonzero monomial degrees of the resulting polynomial remainder, and each coefficient independently certifies its corresponding prime through its additive order. When $n=p_k$ is prime, the least nonzero degree is $p_{k+1}$. Thus the next prime is recovered from the preceding prime alone, without using the index $k$, the prime-counting function, a prime table, nor any primality tests. We develop the underlying ring structure, give equivalent annihilator and quotient formulations, extend the result to shorter intervals, and provide a SageMath implementation.

math.NT↗

arXiv2026

Undecidability, Chaos and Universality in Arithmetic Terms

Arithmetic terms are finite fixed compositions of additions, subtractions, multiplications, divisions with remainder and exponentiations, containing variables interpreted as natural numbers. They build a well-defined notion of closed formula. It is known that every Kalmar elementary function can be expressed as an arithmetic term. In this paper, one studies the power of expression of the arithmetic terms. By interpreting Hilbert's Tenth Problem in arithmetic terms, it is shown that it is undecidable whether one-variable arithmetic term takes the value $0$, or whether two such terms take or not the same values. An algorithm constructs the arithmetic term representing an arbitrary function, which has been defined by a recurrence rule. This construction has various applications. Functions with chaotic behavior, like the Logistic Map, can be expressed as arithmetic terms. Finally, we construct a Turing complete arithmetic term and we express a Turing universal function by an arithmetic term. A somewhat unexpected application: there is a {\it wise} arithmetic term. It gets (the code of) a sentence, (the code of) a formalized theory and a bound $B$, and after performing a constant number of operations, it outputs (the code of) a proof of the sentence using the given theory if such a proof does exist and its length is less than $B$. Otherwise, it outputs $0$.

math.LO↗

arXiv2026

A Minimal Substitution Basis for the Kalmár Elementary Functions

We show that the class of Kalmár elementary functions can be inductively generated from the addition, the integer remainder, and the base-two exponentiation, hence improving previous results by Marchenkov and Mazzanti. We also prove that the substitution basis defined by these three operations is minimal. Furthermore, we discuss alternative substitution bases under arity constraints.

math.LO↗

arXiv2025

Elementary closed-forms for non-trivial divisors

We present several elementary closed-forms that express a non-trivial divisor for every composite integer $n > 1$. Each closed-form consists of a fixed number of elementary arithmetic operations drawn from the set: addition, subtraction, multiplication, integer division, and exponentiation. Two families of closed-forms are developed. First, direct application of the hypercube method yields closed-forms $T_1(n)$, $T_2(n)$, $T_3(n)$, and $T_4(n)$ expressing the smallest prime divisor, largest non-trivial divisor, largest prime divisor, and greatest prime $\leq n$, respectively. The factorial-unwinding technique underlying these hypercube constructions leads to extreme symbolic complexity, motivating our main result: An alternative closed-form $T(n)$ that avoids factorial-unwinding by synthesizing the quadratic residue invariants $χ(n)$ (largest $r$ such that $r^2$ is a divisor) and $ω(n)$ (number of distinct prime divisors) with integer root extraction. Although evaluating these closed-forms requires exponential time, the number of arithmetic operations performed remains constant and independent of the input size $n$. This sharply contrasts with traditional algorithmic methods, where the number of operations required to locate a non-trivial divisor necessarily scales with $n$.

math.NT↗

arXiv2025

On arithmetic terms expressing the prime-counting function and the n-th prime

We present the first fixed-length elementary closed-form expressions for the prime-counting function, $π(n)$, and the $n$-th prime number, $p(n)$. These expressions are arithmetic terms, requiring only a finite and fixed number of elementary arithmetic operations from the set: addition, subtraction, multiplication, integer division, and exponentiation. Mazzanti proved that every Kalmar function can be represented as an arithmetic term. We develop an arithmetic term representing the prime omega function, $ω(n)$, which counts the number of distinct prime divisors of a positive integer $n$. From this term, we find immediately an arithmetic term for the prime-counting function, $π(n)$. Combining these results with a new arithmetic term for binomial coefficients and novel prime-related exponential Diophantine equations, we manage to develop an arithmetic term for the $n$-th prime number, $p(n)$, thereby providing a constructive solution to the fundamental question: Is there an order to the primes?

math.NT↗

arXiv2025

Arithmetic Terms for Multinomial Coefficient Sums

We construct arithmetic terms representing the partial sums of binomial coefficients, and we extend these results to obtain arithmetic terms representing the multisections of binomial coefficient sums. We also introduce an arithmetic term representing a certain type of multinomial coefficient sum and, as an application, we provide an arithmetic term representing the central trinomial coefficients. This solves one of the research problems of the celebrated book Concrete Mathematics, which remained open for nearly thirty years.

math.GM↗

arXiv2025

On modular representations of C-recursive integer sequences

Prunescu and Sauras-Altuzarra showed that all C-recursive sequences of natural numbers have an arithmetic div-mod representation that can be derived from their generating function. This representation consists of computing the quotient of two exponential polynomials and taking the remainder of the result modulo a third exponential polynomial, and works for all integers $n \geq 1$. Using a different approach, Prunescu proved the existence of two other representations, one of which is the mod-mod representation, consisting of two successive remainder computations. This result has two weaknesses: (i) the representation works only ultimately, and (ii) a correction term must be added to the first exponential polynomial. We show that a mod-mod representation without inner correction term holds for all integers $n \geq 1$. This follows directly from the div-mod representation by an arithmetic short-cut from outside.

math.NT↗

arXiv2024

Polynomial quotient rings and Kronecker substitution for deriving combinatorial identities

We introduce a new approach for generating combinatorial identities and formulas by the application of Kronecker substitution to polynomial expansions within quotient rings. Our main result enables the derivation of elementary arithmetic formulas for many C-recursive integer sequences directly from their characteristic polynomials. As sample applications, we present new formulas for the Pell numbers and central binomial coefficients, which are famous integer sequences. These applications lead us to the discovery of a new and unusual formula for the real $n$-th roots of positive integers, $\sqrt[n]{a}$, characterized as the limit of a quotient involving modular exponentiations. From this limit formula we conjecture a fixed-length elementary closed form expression for $\lfloor \sqrt[n]{a} \rfloor$.

math.GM↗

arXiv2024

Elementary Formulas for Greatest Common Divisors and Semiprime Factors

We conjecture new elementary formulas for computing the greatest common divisor (GCD) of two integers, alongside an elementary formula for extracting the prime factors of semiprimes. These formulas are of fixed-length and require only the basic arithmetic operations of: addition, subtraction, multiplication, division with remainder, and exponentiation. Our GCD formulas result from simplifying a formula of Mazzanti and are derived using Kronecker substitution techniques from our earlier research. By applying these GCD formulas together with our recent discovery of an arithmetic expression for $\sqrt{n}$, we are able to derive explicit elementary formulas for the prime factors of a semiprime $n=p q$.

math.GM↗

arXiv2024

A Polynomial Ring Connecting Central Binomial Coefficients and Gould's Sequence

We establish a novel connection between the central binomial coefficients $\binom{2n}{n}$ and Gould's sequence through the construction of a specialized multivariate polynomial quotient ring. Our ring structure is characterized by ideals generated from elements defined by polynomial recurrence relations, and we prove the conditions under which the set of polynomial generators forms a Gröbner basis. By exploring a specific variation of our ring structure, we demonstrate that expanding and evaluating polynomials within the ring yields both the central binomial coefficients and Gould's sequence. Additionally, we present a method for calculating the binomial transforms of these sequences using our ring's unique properties. This work provides new insights into the connections between two fundamental combinatorial sequences and introduces a new tool for integer sequence analysis, with potential applications in number theory and algebraic combinatorics.

math.GM↗