Superpartient

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Superpartient numbers are ratios of the form p/q, where p and q are relatively prime (so that the fraction is reduced to lowest terms), and p - q is greater than 1. In ancient Greece they were called epimeric (epimerēs) ratios, which is literally translated as "above a part." In ancient Greece, fractions like 3/1 and 5/1 were not considered to be epimeric ratios because of their additional restriction that multiples of the fundamental cannot be epimeric. Epimeric ratios were considered to be inferior to epimoric ratios.

All epimeric ratios can be constructed as products of superparticular numbers. This is due to the following useful identity:

[math]\displaystyle \prod_{i \mathop = 1}^{P \mathop - 1} \dfrac {i + 1} {i} = P[/math]

When considering ratios, and particularly when they are ratios for commas, it can be useful to introduce the notion of the degree of epimoricity (not to be confused with epimericity - see link below). In terms of p/q reduced to lowest terms it is p - q. An epimoric ratio has degree 1, the 7-limit comma 245/243 degree 2, the 5-limit comma 128/125 degree 3, and so forth. Størmer's theorem can be extended to show that for each prime limit p and each degree of epimoricity n, there are only finitely many p-limit ratios with degree of epimoricity less than or equal to n.

See Also: ABC, High Quality Commas, and Epimericity