Superpartient ratio
In mathematics, a superpartient ratio, also called an epimeric ratio or a delta-d ratio (d > 1), is a rational number that is greater than 1 and is not superparticular.
More particularly, the ratio takes the form:
- [math]\displaystyle{ \frac{n + d}{n} = 1 + \frac{d}{n} }[/math],
where [math]\displaystyle{ n }[/math] and [math]\displaystyle{ d }[/math] are positive integers, [math]\displaystyle{ d \gt 1 }[/math] and [math]\displaystyle{ d }[/math] is coprime to [math]\displaystyle{ n }[/math].
Etymology
In ancient Greece, they were called epimeric (epimerēs) ratios, which is literally translated as "above a part".
Definitions
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.
Superpartient subcategories
Superpartient ratios can be grouped into subcategories based on the exact difference between the numerator and the denominator. This is known as the degree of epimoricity (not to be confused with epimericity – see link below), or delta (proposed by Kite Giedraitis). This is particularly useful when considering ratios that are commas.
These subcategories are named as delta-2, delta-3, delta-4, etc., or as superbipartient, supertripartient, superquadripartient, etc. Superparticular or epimoric ratios can likewise be named delta-1.
Examples
- Delta-2 (superbipartient) ratios: 3/1, 5/3, 7/5, 9/7, 11/9, 13/11, etc.
- Delta-3 (supertripartient) ratios: 4/1, 5/2, 7/4, 8/5, 10/7, 11/8, etc.
- Delta-4 (superquadripartient) ratios: 5/1, 7/3, 9/5, 11/7, 13/9, 15/11, etc.
Properties
All superpartient ratios can be constructed as products of superparticular numbers. This is due to the following useful identity:
[math]\displaystyle{ \displaystyle \prod_{i \mathop = 1}^{P \mathop - 1} \dfrac {i + 1} {i} = P }[/math]
Størmer's theorem can be extended to show that for each prime limit p and each degree of epimericity n, there are only finitely many p-limit ratios with degree of epimoricity less than or equal to n.