Workshop Recreational Mathematics |
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List of articles |
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I present a small collection of articles adressing some mathematical problems, which I posted in appropriate newsgroups in the usenet. All these articles are of an amateur-mathematician somehow like "recreational mathematics" although I found sometimes practical things. I'm only concerned with rather elementary maths. |
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From the geometric series the hypergeometric series is derived. The limit of these series have a sertain simple formula, which can surprisingly simple be derived with the help of the coefficients of Euler-triangle. Manuscript-level. |
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The main paper is at: last update: 28.5.2004 Hypergeometric Series |
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The problem of the pythagorean triples (mod n) was posed in the newsgroups sci.math, sci.math.research, alt.recreational.mathematic and in the mailing list for seqfans seqfan. In the OEIS there are some related sequences, namely to the problem-power of 2 (x^2 +y^2 = z^2 (mod n)). The sequence is the list of possible solutions for any n from 1 to infinity. I succeeded in finding an emprical formula for the direct (non-counting) calculation of the values for the entries in A(n) (18.12.2003) and I am currently trying to derive that formula analytically. Thus I provide some articles here, which document the state of my observations and derivations. |
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The main paper is at: last update: 15.5.2004 pythagorean triples A list of observations to a generalization for higher exponents of the problem is in last update: 15.5.2004 pythagorean triples: manuscript1 |
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Besides some statistical extrapolations and some sportive records, there is still not much proven in the Collatz-problem, which comes up sometimes in sci.math, alt.math.recreational od de.sci.mathematik. I did some intense studies in the last years, but didn't collect the results in internet-like articles. I'm going to do that time by time. I produced some views into the collatz-tree; some as excel-sheet, some as graphics. A special nice graphic is a fractal like one, which orders the tree into a round brush-like scheme. My last studies concerned the assumed to be easier-to-prove question, whether a loop exists in the 3x+1-problem, or better, how to prove, that that doesn't exist. Several times I supposed, I succeeded, but the proofs were always insufficient (if not false). But they gave new insight in the problem in a way, that I didn't see in the internet anywhere till now. Even if the formulae are still not sufficient to disprove the 3x+1-loop, they illustrate some properties and explain, why a loop in 5x+1 is existent, or give a handy tool, to enumerate the loop candidates and disprove the existence by examination of a finite number of tries. |
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Some graphics last update: 15.5.2004 Collatz Graphs About the impossibility of a loop in 3x+1; arguments towards a proof last update: 15.5.2004 Loop-Discussion 1 |
Gottfried Helms
Univ Kassel