Pure and Applied Mathematics Journal

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Geometric Series of Numbers Approximating Positive Integers

Received: 11 March 2013    Accepted:     Published: 02 April 2013
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Abstract

The predictability of cycles in the series of Pythagorean triples led to an investigation that yielded numbers (x) that are associated with irrational square roots (√n). The cycles recur with geometric factors (cycle factors y) that are comprised of a positive integer x where y = x + √(x^2±1). On raising the cycle factors to the positive integer powers (ym), a series is generated where each consecutive member comes closer and closer to positive integers as the series progresses. A formula associates the square root (√n) with these series. Prime factorising the positive integers in the power series (xm) produces predictable patterns among the prime factors in the series. In general, power series that have each consecutive member in the series come closer to positive integers are limited to (x + √(x^2±r))m where x and r are positive integers and r < (x + 1)2 – x2 for the + r condition and r < x2 – (x – 1)2 for the – r condition.

DOI 10.11648/j.pamj.20130202.15
Published in Pure and Applied Mathematics Journal (Volume 2, Issue 2, April 2013)
Page(s) 79-93
Creative Commons

This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.

Copyright

Copyright © The Author(s), 2024. Published by Science Publishing Group

Keywords

Power Series of Irrational Numbers, Approximating Positive Integers, Factors That Relate Perfect Squares, Prime Factor Patterns

References
[1] MW Bredenkamp, Applied and Computational Mathematics, 2013, 2, 42-53.
[2] All rational right-angled triangles may be raised or reduced to a relatively prime right-angled triangle defined by a pair of positive integer indices (i,j) where i is an uneven number and j is an even number and the even leg (e), the uneven leg (u) and the hypotenuse (h) of the triangle are algebraically defined by the indices (i,j) as follows. u = i2 + ij, e = j2/2 + ij, h = i2 + ij+ j2/2
[3] MW Bredenkamp, Applied and Computational Mathematics, 2013, 2, 36-41.
Author Information
  • Department of Science, Asia-Pacific International University, PO Box 4, MuakLek, Saraburi Province, 18180, Thailand

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    Martin W. Bredenkamp. (2013). Geometric Series of Numbers Approximating Positive Integers. Pure and Applied Mathematics Journal, 2(2), 79-93. https://doi.org/10.11648/j.pamj.20130202.15

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    Martin W. Bredenkamp. Geometric Series of Numbers Approximating Positive Integers. Pure Appl. Math. J. 2013, 2(2), 79-93. doi: 10.11648/j.pamj.20130202.15

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    AMA Style

    Martin W. Bredenkamp. Geometric Series of Numbers Approximating Positive Integers. Pure Appl Math J. 2013;2(2):79-93. doi: 10.11648/j.pamj.20130202.15

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  • @article{10.11648/j.pamj.20130202.15,
      author = {Martin W. Bredenkamp},
      title = {Geometric Series of Numbers Approximating Positive Integers},
      journal = {Pure and Applied Mathematics Journal},
      volume = {2},
      number = {2},
      pages = {79-93},
      doi = {10.11648/j.pamj.20130202.15},
      url = {https://doi.org/10.11648/j.pamj.20130202.15},
      eprint = {https://download.sciencepg.com/pdf/10.11648.j.pamj.20130202.15},
      abstract = {The predictability of cycles in the series of Pythagorean triples led to an investigation that yielded numbers (x) that are associated with irrational square roots (√n). The cycles recur with geometric factors (cycle factors y) that are comprised of a positive integer x where y = x + √(x^2±1). On raising the cycle factors to the positive integer powers (ym), a series is generated where each consecutive member comes closer and closer to positive integers as the series progresses. A formula associates the square root (√n) with these series. Prime factorising the positive integers in the power series (xm) produces predictable patterns among the prime factors in the series. In general, power series that have each consecutive member in the series come closer to positive integers are limited to (x + √(x^2±r))m where x and r are positive integers and r < (x + 1)2 – x2 for the + r condition and r < x2 – (x – 1)2 for the – r condition.},
     year = {2013}
    }
    

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    AB  - The predictability of cycles in the series of Pythagorean triples led to an investigation that yielded numbers (x) that are associated with irrational square roots (√n). The cycles recur with geometric factors (cycle factors y) that are comprised of a positive integer x where y = x + √(x^2±1). On raising the cycle factors to the positive integer powers (ym), a series is generated where each consecutive member comes closer and closer to positive integers as the series progresses. A formula associates the square root (√n) with these series. Prime factorising the positive integers in the power series (xm) produces predictable patterns among the prime factors in the series. In general, power series that have each consecutive member in the series come closer to positive integers are limited to (x + √(x^2±r))m where x and r are positive integers and r < (x + 1)2 – x2 for the + r condition and r < x2 – (x – 1)2 for the – r condition.
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