1 citations · 1 across the 10 of their papers we have counts for
12 papers
On Integer-Division Bases for the Kalmár Elementary Functions
Joseph M. Shunia
Prunescu, Sauras-Altuzarra, and Shunia asked whether addition, totalized integer division, and base-two exponentiation form a substitution basis for the Kalmár elementary functions…
Prime-Interval Algebras
Joseph M. Shunia
Starting from a positive integer and no a priori information about the primes above it, we construct a polynomial quotient ring that recovers exactly the primes in fro…
Undecidability, Chaos and Universality in Arithmetic Terms
Gabriel Istrate, Mihai Prunescu, Joseph M. Shunia
Arithmetic terms are finite fixed compositions of additions, subtractions, multiplications, divisions with remainder and exponentiations, containing variables interpreted as natura…
Elementary closed-forms for non-trivial divisors
Mihai Prunescu, Joseph M. Shunia
We present several elementary closed-forms that express a non-trivial divisor for every composite integer . Each closed-form consists of a fixed number of elementary arithme…
A Minimal Substitution Basis for the Kalmár Elementary Functions
Mihai Prunescu, Lorenzo Sauras-Altuzarra, Joseph M. Shunia
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 previo…
On modular representations of C-recursive integer sequences
Mihai Prunescu, Joseph M. Shunia
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.…
On arithmetic terms expressing the prime-counting function and the n-th prime
Mihai Prunescu, Joseph M. Shunia
We present the first fixed-length elementary closed-form expressions for the prime-counting function, , and the -th prime number, . These expressions are arithmetic…
Arithmetic-term representations for the greatest common divisor
Mihai Prunescu, Joseph Shunia
We construct a new arithmetic-term representation for the function gcd(a,b). As a byproduct, we also deduce a representation gcd(a,b) by a modular term in integer arithmetic.
Elementary Formulas for Greatest Common Divisors and Semiprime Factors
Joseph M. Shunia
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 semiprime…
Polynomial quotient rings and Kronecker substitution for deriving combinatorial identities
Joseph M. Shunia
We introduce a new approach for generating combinatorial identities and formulas by the application of Kronecker substitution to polynomial expansions within quotient rings. Our ma…
A Polynomial Ring Connecting Central Binomial Coefficients and Gould's Sequence
Joseph M. Shunia
We establish a novel connection between the central binomial coefficients and Gould's sequence through the construction of a specialized multivariate polynomial quo…
Arithmetic Terms for Multinomial Coefficient Sums
Joseph M. Shunia, Lorenzo Sauras-Altuzarra
We construct arithmetic terms representing the partial sums of binomial coefficients, and we extend these results to obtain arithmetic terms representing the multisections of binom…