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.
| Published in | Pure and Applied Mathematics Journal (Volume 2, Issue 2) |
| DOI | 10.11648/j.pamj.20130202.15 |
| 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), 2013. Published by Science Publishing Group |
Power Series of Irrational Numbers, Approximating Positive Integers, Factors That Relate Perfect Squares, Prime Factor Patterns
| [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. |
APA Style
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
ACS 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
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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Download@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://article.sciencepublishinggroup.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}
}
TY - JOUR T1 - Geometric Series of Numbers Approximating Positive Integers AU - Martin W. Bredenkamp Y1 - 2013/04/02 PY - 2013 N1 - https://doi.org/10.11648/j.pamj.20130202.15 DO - 10.11648/j.pamj.20130202.15 T2 - Pure and Applied Mathematics Journal JF - Pure and Applied Mathematics Journal JO - Pure and Applied Mathematics Journal SP - 79 EP - 93 PB - Science Publishing Group SN - 2326-9812 UR - https://doi.org/10.11648/j.pamj.20130202.15 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. VL - 2 IS - 2 ER -
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