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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…