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Apr
10
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...
Apr
10
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.
Apr
6
comment Surprising identities / equations
@Neil: very kind -thank you! If I happen to go for it... I'll let you know! :-)
Apr
4
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?
Mar
16
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)?
Mar
15
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)
Mar
12
comment Question about Eremenko's paper on iteration of entire functions
@AlexR: "left-contained" = "self-contained"?
Mar
12
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
Mar
2
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)
Mar
2
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?
Mar
2
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 $a<p$ are difficult and for this there is no easy way (known). An easy example however is $p=11,a=3$ making $11^2|3^5-1$ . Look at "fermat-quotients"; there are tables for everal "brute-force" results online available. Theres also one example known where $p^3 | a^{p-1}-1 $: $113^3 | 68^{112}-1 $
Mar
2
comment Understanding the Power Iterative Method to find eigenvalues
Uhh, good to know this! I've never had this caution...
Feb
28
comment Solving for $x$: $x^x=N$
@zoosuck: didn't find your email adress. But I've put it on my website some time ago. See go.helms-net.de/math/divers/WexZal.pdf (It was a pure ascii-file, so it is not formatted)
Feb
28
comment Solving for $x$: $x^x=N$
If you like some exploring of this, you might have fun reading the article "wexzal" (written in the 90'ies I think) and possibly only accessible via archive.org. When I read it first, it made me really curious. (If you've difficulties to find it online, I've an early copy of it) P.s. another nice treatize on this subject is the "x^x-spindle" which analyzes the behave of the function using a graphical representation.
Feb
26
comment Why formal power series are not considered a system of hypercomplex numbers?
@Anixx: That "LOL" in your comment on the answer of someone who's trying to help you is something indicating I'll better stay away from your questions in the future to avoid to gather such bad loughter myself...
Feb
25
comment If $a\leq (b + c)/2$ with $a,b,c>0$, why $a^2\leq (b^2 + c^2)/2$?
Ah, I missed that, sorry! Let's delete that comment
Feb
22
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.
Feb
21
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? $E_{i=1}^n i = n^{...^{(i+1)^i}} = n^{...^{2^1}}$ ?
Feb
21
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...
Feb
21
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.