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Geometric Series of Numbers Approximating Positive Integers
Pure and Applied Mathematics Journal
Volume 2, Issue 2, April 2013, Pages: 79-93
Received: Mar. 11, 2013; Published: Apr. 2, 2013
Author
Martin W. Bredenkamp, Department of Science, Asia-Pacific International University, PO Box 4, MuakLek, Saraburi Province, 18180, Thailand
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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.
Keywords
Power Series of Irrational Numbers, Approximating Positive Integers, Factors That Relate Perfect Squares, Prime Factor Patterns
Martin W. Bredenkamp, Geometric Series of Numbers Approximating Positive Integers, Pure and Applied Mathematics Journal. Vol. 2, No. 2, 2013, pp. 79-93. doi: 10.11648/j.pamj.20130202.15
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.
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