Friday, February 4, 2011

Applications and examples

Applications and examples of experimental mathematics include:
  • Searching for a counterexample to a conjecture
    • Roger Frye used experimental mathematics techniques to find the smallest counterexample to Euler's sum of powers conjecture.
    • The ZetaGrid project was set up to search for a counterexample to the Riemann hypothesis.
    • This project is searching for a counterexample to the Collatz conjecture.
  • Finding new examples of numbers or objects with particular properties
    • The Great Internet Mersenne Prime Search is searching for new Mersenne primes.
    • The distributed.net's OGR project is searching for optimal Golomb rulers.
    • The Riesel Sieve project is searching for the smallest Riesel number.
    • The Seventeen or Bust project is searching for the smallest Sierpinski number.
    • The Sudoku Project is searching for a solution to the minimum Sudoku problem.
  • Finding serendipitous numerical patterns
    • Edward Lorenz found the Lorenz attractor, an early example of a chaotic dynamical system, by investigating anomalous behaviours in a numerical weather model.
    • The Ulam spiral was discovered by accident.
    • Mitchell Feigenbaum's discovery of the Feigenbaum constant was based initially on numerical observations, followed by a rigorous proof.
  • Use of computer programs to check a large but finite number of cases to complete a computer-assisted proof by exhaustion
    • Thomas Hales's proof of the Kepler conjecture.
    • Various proofs of the four colour theorem.
    • Clement Lam's proof of the non-existence of a finite projective plane of order 10.
  • Symbolic validation (via Computer algebra) of conjectures to motivate the search for an analytical proof
    • Solutions to a special case of the quantum three-body problem known as the hydrogen molecule-ion were found standard quantum chemistry basis sets before realizing they all lead to the same unique analytical solution in terms of a generalization of the Lambert W function. Related to this work is the isolation of a previously unknown link between gravity theory and quantum mechanics in lower dimensions (see quantum gravity and references therein).
    • In the realm of relativistic many-bodied mechanics, namely the time-symmetric Wheeler–Feynman absorber theory: the equivalence between an advanced Liénard–Wiechert potential of particle j acting on particle i and the corresponding potential for particle i acting on particle j was demonstrated exhaustively to order 1 / c10 before being proved mathematically.
  • Evaluation of infinite series, infinite products and integrals (also see symbolic integration), typically by carrying out a high precision numerical calculation, and then using an integer relation algorithm (such as the Inverse Symbolic Calculator) to find a linear combination of mathematical constants that matches this value. For example, the following identity was first conjectured by Enrico Au-Yeung, a student of Jonathan Borwein using computer search and PSLQ algorithm in 1993:
\begin{align}\sum_{k=1}^\infty \frac{1}{k^2}\left(1+\frac{1}{2}+\frac{1}{3}+\cdots+\frac{1}{k}\right)^2 = \frac{17\pi^4}{360}.\end{align}
  • Visual investigations
    • In Indra's Pearls David Mumford and others investigated various properties of Möbius transformation and Schottky group using computer generated images of the groups which: furnished convincing evidence for many conjectures and lures to further exploration.

Open problems

Some relations have been shown to hold to very high precision, but no formal proof has yet been found; one example is:
\begin{align}\sum_{n=0}^\infty \left(\frac{1}{(7n+1)^2}+\frac{1}{(7n+2)^2}-\frac{1}{(7n+3)^2}+\frac{1}{(7n+4)^2}-\frac{1}{(7n+5)^2}-\frac{1}{(7n+6)^2}\right)\approx\frac{24}{7\sqrt{7}}\int_{\pi/3}^{\pi/2} \log \left| \frac{\tan t + \sqrt{7} }{\tan t - \sqrt{7} } \right|dt\end{align} \, .

Plausible but false examples

Main article: mathematical coincidence
Some plausible relations hold to a high degree of accuracy, but are still not true. One example is:
\int_{0}^{\infty}\cos(2x)\prod_{n=1}^{\infty}\cos\left(\frac{x}{n}\right)dx \approx \frac{\pi}{8}.
The two sides of this expression only differ after the 42nd decimal place.
Another example is that the maximum height (maximum absolute value of coefficients) of all the factors of xn − 1 appears to be the same as height of nth cyclotomic polynomial. This was shown by computer to be true for n < 10000 and was expected to be true for all n. However, a larger computer search showed that this equality fails to hold for n = 14235, when the height of the nth cyclotomic polynomial is 2, but maximum height of the factors is 3.
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