Overview
When considering relations between constants, sometimes a slight change in perspective can dramatically simplify a relation. This can not only reduce the relation’s complexity, but possibly even provide better insight into the relation, especially in the case that it is only a conjecture, in which case this can provide a clue for proving it.
Background
Given a vector of real numbers \(v=[x_1,x_2,…,x_n]\in\mathbb{R}^n\), a relation on it is defined as a multivariate polynomial \(p\) with integer coefficients such that \(p(v)=0\). In general, \(p\) can be written using multi-index notation:
\(\displaystyle p(x)=\sum_{\alpha\in\mathbb{N}_0^n}{a_\alpha x^\alpha}=\sum_{\alpha\in\mathbb{N}_0^n}{a_{(\alpha_1,\dots,\alpha_n)} x_1^{\alpha_1} x_2^{\alpha_2}\cdots x_n^{\alpha_n}}\)
Then, one can define the degree of \(p\) to be \(\max{\{\|\alpha\|_1\mid a_\alpha\neq 0\}}\) and the order of \(p\) to be \(\max{\{\|\alpha\|_\infty\mid a_\alpha\neq 0\}}\). In other words, the degree is defined by the term with the largest sum of exponents, and the order is the largest single exponent that appears in any of the terms. Then, when \(v\) has a polynomial relaion with degree \(d\) and order \(o\), it is said that the constants in \(v\) have a relation of degree \(d\) and order \(o\).
The simplest examples of such relations have order \(1\). Then, degree \(1\) implies affine relations, such as between \(\sqrt{5}\) and the golden ratio \(\phi\): Since \(\phi=\frac{1+\sqrt{5}}{2}\), one may equivalently write this as the relation \(2\phi-\sqrt{5}-1=0\).
Another simple example is degree \(2\) and order \(1\), in which case one constant may be extracted as a möbius transformation of the others. This includes all of the möbius relations found by the Ramanujan team, such as:
\(\frac{e+1}{e-1}=\mathrm{PCF}[4n+2,1]=2+\frac{1}{6+\frac{1}{10+\dots}}:=P_1\)
Or equivalently, \(P_1 e-P_1-e-1=0\). Note here the product \(P_1 e\) showing that the degree is greater than \(1\).
Motivation
Relations with order \(o>1\) can get cumbersome very quickly. Consider, for instance, the following formula by Ramanujan:
This formula can be naïvely transformed into a relation of four constants (using Euler’s continued fraction formula on the sum):
\(\pi,\, e,\, P_1:=\mathrm{PCF}[1,n],\, P_2:=\mathrm{PCF}[2+2n,1-2n]\)
This relation has degree \(6\) and order \(2\):
\(\pi e P_1^2 P_2^2 \,-\, 2\pi e P_1^2 P_2 \,+\, \pi e P_1^2 \,-\, 2P_2^2 \,+\, 4P_2 \,-\, 2 \,-\, 4P_1 P_2 \,+\, 4P_1 \,-\, 2P_1^2 \,=\, 0\)
However, the original formulation by Ramanujan transforms much more cleanly to a möbius transformation of \(P_1,\,P_2\) each:
\(\sqrt{\frac{\pi e}{2}}=\frac{1}{P_1}+\frac{1}{P_2-1}\)
And then squaring both sides yields a mere degree \(2\) order \(2\) relation on \(\pi,\,e,\,\frac{1}{P_1}\) and \(\frac{1}{P_2-1}\):
\(\pi e \,-\, 2(\frac{1}{P_1})^2 \,-\, 4\frac{1}{P_1}\frac{1}{P_2-1} \,-\, 2(\frac{1}{P_2-1})^2 \,=\, 0\)
The implication of this example is that choosing the four constants “properly”, in this case to be \(\pi,\,e,\,\frac{1}{P_1},\,\frac{1}{P_2-1}\) instead of \(\pi,\,e,\,P_1,\,P_2\), can dramatically simplify the relation.
The challenge
We want to develop an algorithm the identifies a best representation of the constants, or at least approximates it:
Given a general relation of order \(o>1\), the algorithm will find for each constant a möbius transform with integer coefficients, such that the transformed constants have a relation of a lower degree (and maybe even a lower order). That is, it finds the möbius transforms on the constants that minimize (or at least reduce) the degree of the relation.
Given that such an algorithm has been developed and is capable of simplifying relations in the way we described, applying it to new, more complex relations generated from our other search algorithms can speed up the process of proving the conjectures we generate (or otherwise disproving them).