Computational considerations on the representation of number-theoretic functions by arithmetic terms.

This article focuses on constructing explicit closed-form arithmetic-term representations for several fundamental number-theoretic functions, including the p-adic valuation, number-of-divisors function (τ), sum-of-divisors function (σ), Euler’s totient function (φ), modular inverse, integer parts of roots and logarithms, multiplicative order, and discrete logarithm. Building on and extending methods related to Hilbert’s Tenth Problem and Kalmar functions, the authors develop the hypercube method, which transforms exponential Diophantine equations defining these functions into arithmetic terms involving only elementary operations, the Hamming weight, and generalized geometric progressions up to the second kind. Although these arithmetic terms are typically very large and computationally intensive, the paper provides constructive proofs and Maple code implementations that verify their correctness for small inputs, thereby demonstrating the existence and explicit construction of such closed forms without relying on prime factorizations or iterative algorithms. The work also discusses logical definability aspects and offers a framework encouraging further research into shorter or more efficient arithmetic-term representations.