Algorithm-assisted discovery of an intrinsic order among mathematical constants

Area of Science:

• Mathematics
• Computational Science
• Number Theory

Background:

• Computer algorithms increasingly assist mathematical discoveries, particularly in exploring vast parameter spaces.
• The synergy between human intuition and computational power offers potential for uncovering elusive mathematical structures.

Purpose of the Study:

• To demonstrate computer-assisted discovery of a novel mathematical structure: the conservative matrix field.
• To explore the potential of computational approaches in finding new mathematical formulas and connections.

Main Methods:

• Developed a massively parallel computer algorithm, inspired by the Ramanujan Machine project, to generate numerous continued fraction formulas for mathematical constants.
• Analyzed patterns in the generated formulas to construct and characterize conservative matrix fields.
• Utilized a distributed computing network of thousands of personal computers.

Main Results:

• Discovered and constructed the first conservative matrix fields, revealing unexpected relationships between mathematical constants like π, ln(2), e, and the Gompertz constant.
• Demonstrated the ability of conservative matrix fields to unify hundreds of existing formulas and generate infinite new ones.
• Applied matrix fields to values of the Riemann zeta function (ζ(n)) and generalized Apéry's proof for the irrationality of ζ(3).

Conclusions:

• Conservative matrix fields represent a significant new mathematical structure with broad implications across mathematics and physics.
• Large-scale computational approaches are powerful tools for addressing longstanding mathematical problems and uncovering novel scientific connections.
• This work highlights the potential of combining computational power with mathematical insight for future discoveries.