This paper introduces the Arya Invariant, a new digit-level classification of positive integers based on the prime factorization of individual decimal digits. For each decimal digit d, we define its Factor Pair Sum fps(d) as the sum of its prime factors counted with multiplicity, and its Arya value as QFS(d) = d − fps(d). A positive integer is called Arya-balanced if the Arya values of all its digits sum to zero. The central result is the Arya Digit Sum Identity: a number is Arya-balanced if and only if its ordinary digit sum equals its factor-pair digit sum. A complete elementary algebraic proof is provided and verified computationally over all integers from 1 to 100,000, finding 12,831 Arya-balanced numbers with zero failures. The paper additionally presents two computational observations: the structure of consecutive Arya-balanced pairs (Twin Balanced Numbers), and the use of Arya-balanced frequency as a statistical test for digit-level bias in pseudorandom number generators — a bias undetectable by standard bit-frequency tests. Four open problems are stated for future investigation.
Aryaveer Sharma (Sun,) studied this question.