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Experimental Mathematics

Introduction

• Recent years have experienced growth in experimental mathematics, utilizing modern computational tools for mathematical research.

• This practice is widespread, involving many researchers across various fields, including number theory, algebra, analysis, geometry, and topology.

• Key tasks include generating examples, creating plots, manipulating algebra, testing conjectures, and exploring proofs.

• Computation aids mathematicians by providing time-efficient conjecture testing, largely due to the development of advanced symbolic mathematics software.

• Significant tools:

  • Maple and Mathematica: Leading symbolic mathematics software.

  • Magma: Computational algebra package from the University of Sydney.

  • Other tools for integer sequences and constants.

Historical Context

• Gordon Moore's prediction in 1965 about hardware complexity has shown consistent accuracy, leading to unprecedented hardware advancements.

• Experimental Mathematics Journal: Established 12 years ago, reflects this shift in mathematical research perspective.

• Past perception that “real mathematicians don’t compute” has evolved significantly, with extensive computational papers contributing to mainstream mathematics.

Key Methods in Experimental Mathematics

Integer Relation Detection

• Definition: Searches for linear relationships satisfied by a set of numerical values represented by vectors.

• Integer relation algorithm finds integer coefficients such that a linear combination of the vector equals zero.

• Requires high precision and can lead to significant computational requirements; for instance, a 50,000-digit calculation was necessary for a recent result.

• The PSLQ algorithm: The best-known method for detecting integer relations, recognized for its efficiency.

Arbitrary Digit Calculation Formulas

• A notable application of PSLQ: The BBP formula for π:

  • Allows for the calculation of binary or hexadecimal digits of π without computing preceding digits, conserving memory.

  • Example of similar calculations includes the binary expansion for log 2.

• The BBP approach has led to various other formulas allowing for arbitrary-position digit calculations for different constants.

Discoveries and Implications

Findings on π and Normality

• Despite efforts, no BBP-type formula for π exists in bases other than powers of two.

• Recent findings indicate the computational exploration has significant implications for the normality of mathematical constants, including π and log 2.

• Definitions:

  • b-normal: A real number is b-normal if every possible string of digits in base b appears with a limiting frequency.

  • Many known constants are suspected to be normal, but proven examples are few.

Euler’s Multi-Zeta Sums

• Multi-zeta constants generalize the Riemann zeta function and are crucial for modern mathematical evaluations.

• Calculations of complex sums lead to interesting numerical identities, heuristically verified through computer experiments instead of traditional proof methods.

Integral Evaluation Techniques

• Advanced numerical methods offer high-precision results, often exceeding typical software limits.

• Innovations such as tanh-sinh quadrature significantly improve integration accuracy over traditional methods, revealing new mathematical patterns and facilitating research implementations.

Ramanujan's Continued Fractions

• Ramanujan discovered noteworthy relationships involving continued fractions, experimentally leading to conclusions about logarithmic constants in mathematics.

Conclusion

• The intersection of computational experimentation and traditional mathematical research is increasingly prominent.

• Future advances in software and hardware promise further impact on research methodologies and findings.

• Ongoing exploration raises questions about rigorous proof versus computational evidence, reaffirming the necessity of both in the mathematical community.