Visible Structures in Number Theory

Creates visualizations that are similar to some of the ones that appear in "Visible Structures in Number Theory" by Peter Borwein and Loki Jorgensen (December 2001 American Mathematical Monthly, pages 897-910). Notes on the representations.

The basic idea is to start by representing a positive real number by a sequence of positive integers. The first k2, k = 20, 40, 100, or 250, of these numbers are then displayed mod m. These mod m integers are displayed starting at the top left, proceeding row by row.

  • Number to be represented:
  • Representation:
    • Digits
      Base:
    • Continued Fraction (Display size currently max's out at "Medium")
    • Integer parts of multiples.
  • Modulus:
  • Display size:
    • - Small (20 by 20)
    • - Medium (40 by 40)
    • - Large (100 by 100)
    • - Huge (250 by 250)
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Notes on the representations

  1. Digits - the digits of the base b representation of the number.
  2. Continued Fraction - The simple continued fraction of a number x is a list that has as its first entry [x], the integer part of x. The list ends right there if x is an integer; and if not, the rest of the list is the simple continued fraction representation of 1/(x-[x]). For examples, see a related webMathematica page. Right now, the continued fraction representations of some numbers get truncated. I think it's a restriction on this account, but I'm not sure.
  3. Integer parts of multiples - A fairly simple idea: take the integer parts of the multiples x, 2x, 3x, ...; and then reduce mod m. Do these sequences uniquely characterize x?
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Ken Levasseur
Mathematical Sciences
UMass Lowell
Kenneth_Levasseur@uml.edu
Ken's WebMathematica Scripts

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