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May
14
comment Algorithm for tetration to work with floating point numbers
It were tetration if you could find some meaningful solution for the idea, that the "i++" in the last loop could be fractional instead of "i=i+1" ...
May
11
comment The Physical Meaning of Tetration with fractional power tower
@MphLee: I find your comment well written, and the keyword "universe" reminds me of some (so far fruitless) speculation, whether the problem of the hypothesized extreme dilatation in the near of the big-bang might be a candidate for mathematical models involving iterated exponentiation . A second, even more, speculative idea was when the evolution of the time itself was discussed. Again this has for me some flavor of selfcomposed functionality. There is also an electronic tool, the "avalange diode". Perhaps its curve is already perfectly modeled- otherwise that would be a candidate to look at.
May
10
comment Continuum between addition, multiplication and exponentiation?
@DanW: yes, I was really impressed. Thanks for the reminding!
May
9
comment Continuum between addition, multiplication and exponentiation?
@Daniel: yes; in the first example I work with $t=1.01$ which is also the (lower) fixpoint of $b=t^{1/t} \approx 1.019$ and attracting for the exponentiation, because the upper fixpoint is above $10$ which is the limit of the the intended multiplication table. In the second example I think the value for the base is not so important and might go up to $b=4$ or $b=10$ (although I didn't really look at it)
May
8
comment Is the sum of the reciprocals of the squarefree numbers divergent or convergent?
@Darth: thanks too - that formal re-expression of the sum is a nice reminder to look at - perhaps it helps to understand things better if needed later.
May
8
comment Is the sum of the reciprocals of the squarefree numbers divergent or convergent?
@A.P. : <arrgh> reading your argument makes me feeling nearly stupid (but well, it's ok). Of course! (I asked question Q2 there more out of couriosity.... )
Apr
26
comment Closed Form Summation Example
Hmm, wouldn't it be clearer to write $ \displaystyle a \cdot (\sum_{i=1}^n i) $ and $\displaystyle b \cdot (\sum_{i=1}^n 1)$ ? The closed-forms for the sum-expressions should even easier jump out from the memory...
Apr
25
comment Longest known sequence of identical consecutive Collatz sequence lengths?
@Pure : yes I've seen that. But I've only temporarily time, due to familiar duties...
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)