Conway's subprime Fibonacci sequences

It's the age-old recurrence with a twist: sum the last two terms and if the result is composite, divide by its smallest prime divisor to get the next term (e.g., 0, 1, 1, 2, 3, 5, 4, 3, 7, ...). These sequences exhibit pseudo-random behaviour and generally terminate in a handful of cycles, properties reminiscent of 3x+1 and related sequences. We examine the elementary properties of these 'subprime' Fibonacci sequences.Comment: 18 pages, 5 figure

The Yellowstone Permutation

Define a sequence of positive integers by the rule that a(n) = n for 1 <= n = 4, a(n) is the smallest number not already in the sequence which has a common factor with a(n-2) and is relatively prime to a(n-1). We show that this is a permutation of the positive integers. The remarkable graph of this sequence consists of runs of alternating even and odd numbers, interrupted by small downward spikes followed by large upward spikes, suggesting the eruption of geysers in Yellowstone National Park. On a larger scale the points appear to lie on infinitely many distinct curves. There are several unanswered questions concerning the locations of these spikes and the equations for these curves.Comment: 10 pages, 6 figures. Mar 7 2015: mostly stylistic change

On integer linear combinations of terms of rational cycles for the generalized 3x+1 problem

In the paper, some special linear combinations of the terms of rational cycles of generalized Collatz sequences are studied. It is proved that if the coefficients of the linear combinations satisfy some conditions then these linear combinations are integers. The discussed results are demonstrated on some examples. In some particular cases the obtained results can be used to explain some patterns of digits in $p$-adic representations of the terms of the rational cycles

3x+13x+1 inverse orbit generating functions almost always have natural boundaries

The $3x+k$ function $T_{k}(n)$ sends $n$ to $(3n+k)/2$ resp. $n/2,$ according as $n$ is odd, resp. even, where $k \equiv \pm 1~(\bmod \, 6)$. The map $T_k(\cdot)$ sends integers to integers, and for $m \ge 1$ let $n \rightarrow m$ mean that $m$ is in the forward orbit of $n$ under iteration of $T_k(\cdot).$ We consider the generating functions $f_{k,m}(z) = \sum_{n>0, n \rightarrow m} z^{n},$ which are holomorphic in the unit disk. We give sufficient conditions on $(k,m)$ for the functions $f_{k, m}(z)$ have the unit circle $\{|z|=1\}$ as a natural boundary to analytic continuation. For the $3x+1$ function these conditions hold for all $m \ge 1$ to show that $f_{1,m}(z)$ has the unit circle as a natural boundary except possibly for $m= 1, 2, 4$ and $8$. The $3x+1$ Conjecture is equivalent to the assertion that $f_{1, m}(z)$ is a rational function of $z$ for the remaining values $m=1,2, 4, 8$.Comment: 15 page

The Long Search for Collatz Counterexamples

Despite decades of effort, the Collatz conjecture remains neither proved, nor refuted by a counterexample, nor formally shown to be undecidable. This note introduces the Collatz problem and probes its logical depth with a test question: can the search space for counterexamples be iteratively reduced, and when would it help

Fixation and stabilization of heavy metals in a product composed of fly ash and industrial acidic wastes

Thesis (Doctoral)--Izmir Institute of Technology, Chemical Engineering, Izmir, 2010Includes bibliographical references (leaves: 137-149)Text in English; Abstract: Turkish and Englishxvii, 149 leavesLarge quantities of fly ash are produced in the world while only a very small portion of it finds utilization as a product. Since even larger quantities of fly ash should be expected to be generated in an increasingly energy-hungry world, finding new, smart pathways towards fly ash utilization carries its own weight in achieving sustainable development. Manufacturing light-weight fly ash aggregates which could also be used as sinks for other wastes with acceptable environmental foot-print and possible added value is exactly one such pathway. This thesis comprises of the studies where a fly ash sample from a large Thermal Power Plant (Soma fly ash) is used to stabilize an extremely acidic industrial waste in an aggregate structure with the aim that the aggregates produced had acceptable environmental impact and possible product value. The study is divided into several parts. Initial chapters contain the characterization studies of both the fly ash from Soma, Manisa Thermal Power Plant and the highly acidic industrial waste from a metal stripping factory in Cigli-Izmir. Later chapters contain the methodology and the experimental parts of the work. The acidity of the aggregates was also optimized keeping in mind the present environmental regulations. Solid/liquid ratio, effect of aging of the aggregates, effect of leaching from the aggregates and their mechanical strengths were the main experimental parameters investigated. Mineralogical composition of the aggregates was compared with the raw fly ash to shed light on the fixation mechanism. The fly ash was rich in especially Si, Al and Ca and some heavy metals in trace amounts whereas the acidic waste contained huge amounts of Zn, Fe and other heavy metals. This study demonstrated conclusively that the acidic waste was not only neutralized by the fly ash to disposable levels, but also that the leaching of both the major and minor elements from the aggregate body into the water was below the environmental limits in force both in Europe and Turkey today. The produced aggregates were also observed to possess the low-strength-material properties which open the possibility for their utilization in construction applications