Research Article
On the Discrete and Continuous Harmonic Encoding of Primes
Ioannis Papadakis*
Issue:
Volume 11, Issue 3, June 2026
Pages:
45-56
Received:
4 April 2026
Accepted:
15 April 2026
Published:
21 May 2026
DOI:
10.11648/j.mcs.20261103.11
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Abstract: The dichotomy between the prime-based multiplicative and additive representations of integers poses fundamental and distinct challenges in analyzing the underlying prime distribution. This paper contributes to this area of research by introducing a tripartite framework for the lossless mathematical encoding of primes and their additive partitions. First, we establish Hybrid Prime Factorization (HPF) as a bounded a priori structural prime-generation framework. On certified intervals, primality is forced by disjoint partitions of canonical prime bases under the stated magnitude bound, so that certain HPF configurations yield prime outputs structurally, i.e., without requiring a separate post hoc primality test of the evaluated output. Second, to address the linear expansion of additive partitions, we introduce a deterministic pairing map, L(N), which losslessly
compresses the entire additive state of even integer partitions into a single, uniquely factorizable scalar in ℤ. Finally, recognizing the asymptotically factorial limits of discrete integer representation, we map this arithmetic complexity into the
continuous domain. We derive bounded, piecewise smooth harmonic sieve functions over ℝ \ ℤ that isolate prime and composite structures through the limits of indeterminate trigonometric forms. This progression establishes that prime complexity need not be confined to discrete combinatorial bounds, but can be translated into continuous harmonic functions, demonstrating that the prime counting function π(x) can be generated as a sum of continuous harmonic trigonometric functions.
Abstract: The dichotomy between the prime-based multiplicative and additive representations of integers poses fundamental and distinct challenges in analyzing the underlying prime distribution. This paper contributes to this area of research by introducing a tripartite framework for the lossless mathematical encoding of primes and their additive partitions. ...
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