Periodicity and Number Theory
The Farey sequence, as defined in number theory, provides a modular classification of entities made up of integral numbers of two types of particle, such as protons and neutrons. If the stability of such entities is assumed to depend on an increasing excess of one particle type (neutrons), compositions (nuclides) of constant excess are shown to be stabilized over limited regions, related in extent to the golden ratio, and leading to a periodic relationship that depends on relative stabilities. This stability trend is shown to be identical to the hypothetical periodicity amongst stable nuclides, postulated before on the basis of prime-number distribution on a spiral. Triangles of stability that limit the number of possible nuclides are shown to derive from limiting ratios, defined by fractions generated by the Farey procedure from Fibonacci numbers. The results correlate well with experimental stabilities inferred from measured mass defects and with solar abundances.
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