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When has numerology been successfully used in math and science? The Monstrous Moonshine conjecture led to a Fields medal for Borcherds. Balmer's formula for hydrogen spectra led to the Bohr model of the atom.

We could extend this to general computational experimentation. For example, the Birch-Swinnerton-Dyer conjecture was originally formulated based on sketchy computational results. Gauss guessed the law of quadratic reciprocity and the prime number theorem from his calculations too. Are there other interesting or instructive examples?

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    $\begingroup$ I would say that most conjectures in number theory are supported by computational experimentation. What do you mean by "numerology"? $\endgroup$
    – Pete L. Clark
    Commented Jun 18, 2010 at 18:40
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    $\begingroup$ I think "numerology" here means conjectures that are supported by less computational evidence than someone with ridiculously high standards would like. $\endgroup$
    – Michael Lugo
    Commented Jun 18, 2010 at 18:43
  • $\begingroup$ Mirror symmetry? $\endgroup$
    – Steve Huntsman
    Commented Jun 18, 2010 at 18:56
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    $\begingroup$ Oh, and PSLQ: en.wikipedia.org/wiki/Integer_relation_algorithm#Applications $\endgroup$
    – Steve Huntsman
    Commented Jun 18, 2010 at 19:30

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Three examples from the theory of dynamical systems (in a broad sense).

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Searching for "experimental mathematics" will give you entire books and a journal's worth of examples. This seems to be too broad a question for MathOverflow. However, I will give one example: The Riemann hypothesis. In Borwein and Bailey's book they give evidence that Riemann arrived at this conjecture by means of calculating the first few zeros.

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There is this MO thread Experimental Mathematics about experimental mathematics, with more than 40 answers.

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  • $\begingroup$ Good catch...maybe this question should be closed as a duplicate? $\endgroup$
    – Timothy Chow
    Commented Jun 18, 2010 at 23:14
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Perhaps it would be useful to record some of the failures of numerology as well, e.g., Kepler's attempt to model the solar system by inserting the five Platonic solids among the six planets that were known in his day. Kepler's eventual success with ellipses could be viewed as numerology as well, as the theoretical basis for it had to wait for Newton.

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  • $\begingroup$ Kepler is a very interesting case, because his work relied almost entirely on experimentation (of good, scientific kind) $\textit{and}$ he sometimes invoked astrological and numerological ideas. For example, in spite of correctly concluding that it is sectorial area, not arc length, that describes the motion of planets, he went on to "prove" that the circumference of the ellipse is $\pi(a+b).$ $\endgroup$
    – Victor Protsak
    Commented Jun 18, 2010 at 23:21
  • $\begingroup$ He claims a false relation between the volume of a "lemon" and a spherical segment formed by rotating a circular segment around, respectively, the chord and the axis of symmetry perpendicular to it, which he numerically confirms for a special segment of radius 100000. Gulden, one of his critics, acerbically commented that Kepler advises others to find a legitimate demonstration... appealing to numbers... so let the numbers decide whether or not one should search for something that does not exist, and goes on to trash Kepler's conclusion using a segment with radius 10000000. $\endgroup$
    – Victor Protsak
    Commented Jun 18, 2010 at 23:30
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    $\begingroup$ However, one cannot discount even his more speculative treatises: "On six-cornered snowflakes" earned him the reputation of the founder of crystallography; some figures from "Harmonices Mundi" have been seen as representing the first examples of aperiodic tilings. $\endgroup$
    – Victor Protsak
    Commented Jun 18, 2010 at 23:40

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