Gottfried Helms
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 Apr10 comment Given $\log 2$ and $\log 3$, compute $\log 120$ ? and who does tell the OP that the base-10-log was meant? I think one should find out where that precondition was implicitely hidden and how it could be pointed out to the OP... Apr10 comment Inverse and named fixed values, with ↑↑? My proposal is to use "height" from the idea, that the integer version of the tetration is often understood as "power tower" (right associative) and is also derived from number of iterations, so "height" might be the most useful name, generalizable in the context of hyperoperations "iterative height of exponentiation", "of multiplication" etc. In my software I use thus $\operatorname{hgh}(x)$ for this and $\operatorname{hgh}(x_1,x_0)$ if I want precisely express the iteration-"height" from one $x_0$ to another $x_1$ by the (generalized) iteration of the current operation under consideration. Apr6 comment Surprising identities / equations @Neil: very kind -thank you! If I happen to go for it... I'll let you know! :-) Apr4 comment A question about the $3n+1$ conjecture 1) I like the way you explained the last two collatz-questions... 2) I tried to write the same logic for some other cases. For instance, ${ 3(n-1)+1 \over 4} = \frac 34 (n-1)+1$ : your example recurs to the property, that $n=-1$ generates a cycle, I try to use the same logic derived from $n=+1$ because also this gives a cycle. Then another case is $n=-5 \to n=-7 \to n=-5$ giving another cycle, so your form adapted ${ 3(n+5)+1 \over 2 }$ should give a similar nice rule. But perhaps I'm still to sleepy to formulate this correctly. Do you see, how this could be made an analoguous statement? Mar16 comment Example for $\dfrac{p_1p_2-1}{p_1+p_2}$ being odd natural number . But how can with that examples the result be an odd number (as required in the OP)? Mar15 comment Graham's number of layer 1 tetration explanation? A discussion of this can also be found at mpmueller.net/reihenalgebra.pdf (However I don't know whether this is really helpful for you, since your notations seem to be very similar to that what I remember from an early version of the article) Mar12 comment Question about Eremenko's paper on iteration of entire functions @AlexR: "left-contained" = "self-contained"? Mar12 comment What is the name of this irrational math constant and is there a compact way to write it? 0.10110111011110… Subtract it from 1/9 = 0.1111111... and express the resulting pattern of ones by the according negative powers of 10 Mar2 comment How do I find the Cesaro sum of the series $\{1, -1, 1, -1, …\}$? Very nice! Thank you for kindness! (I'll delete then my comment) Mar2 comment How do I find the Cesaro sum of the series $\{1, -1, 1, -1, …\}$? So this helps the OP to understand the Cesaro-summation? Mar2 comment Is $2^{1093}-2$ a multiple of $1093^2$? For the last remark: cases $a>p$ can be generated to arbitrary powers of $p$; but the cases $a0$, why $a^2\leq (b^2 + c^2)/2$? Ah, I missed that, sorry! Let's delete that comment Feb22 comment Is anything known about the size of the smallest number with stopping time $n$ Hmm, while I'm not yet getting the meaning of the $c_k(n)=a_k^{(n)}n...$ formula - perhaps you find my discussion on go.helms-net.de/math/collatz/Collatz061102.pdf inspiring where I propose and discuss the problem using the view from vectors of exponents and find some criterion-formulae for the problem of existence of cycles. Feb21 comment Is there a notation for exponentiation analog to capital-sigma notation Σ for addition and capital pi Π notation for multiplications? After it is obvious that there is no analoguous notation in use, why not simply propose to use big $E$ for this? $E_{i=1}^n i = n^{...^{(i+1)^i}} = n^{...^{2^1}}$ ? Feb21 comment Is anything known about the size of the smallest number with stopping time $n$ Interesting! I denote the basic Collatz-transformation as one step $a_{k+1}=(3a_k+1)/2^{A_k}$ which is on the odd integers only (and thus determines the exponent $A_k$ exactly). Then I look at a segment of the trajectories as by a vector of that exponents only: $a_k=T(a_0; A_0, A_1,A_2,...,A_{k-1})$. Then -leaving even $a_0$ indeterminate- I can easily form such vectors of exponents of any length such that intermediate $a_k \gt a_0$. Implementing stopping times of any length - and then find $a_0,a_k$ by looking at the modular condition. Unfortunately this does not allow to find records... Feb21 comment Is anything known about the size of the smallest number with stopping time $n$ @user252...: hmm, I think Tomas Oliveira should already have the most efficient algorithms because he is also looking for stopping-time records and the sort. I've never much engaged in that point of view (of stopping time and records); I was always involved in the analytical/modular aspects of the problem especially the problem of existence of cycles.