Ben-Gen: A Symbolic Pattern Language for Representing Astronomically Large Numbers

Abstract

This paper introduces Ben-Gen, a symbolic pattern language designed to concisely represent integers with highly regular digit structures, particularly those exhibiting exponential growth or zero-dominant patterns. Unlike general-purpose arithmetic systems or data compression algorithms, Ben-Gen does not aim to encode arbitrary numerical data. Instead, it provides a formal notation and grammar for describing specific classes of structured numbers in a compact and human-readable form. We present the syntax, semantics, and axioms underlying Ben-Gen, with illustrative examples and step-by-step expansions. The limitations of the system are explicitly discussed, including its inapplicability to random data, cryptographic use, or lossless compression. Potential applications include mathematics education, visualization of large numbers, and procedural generation of structured numerical data. This work is intended as a foundational and exploratory contribution toward symbolic representations of large numerical patterns. Full formal definitions, derivations, and additional examples are available in the accompanying manuscript or upon request, providing a complete framework for researchers and educators interested in structured numerical representations.

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