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8h
comment Integration of a polylogarithm: Is this function known?
Is this an error? I've never seen two times $dx$ in one integration formula
22h
comment Prove that $24^{31}$ is congruent to $23^{32}$ mod 19.
To do your last formula mentally it seems more practical to reorder: $(5 \cdot 4) \cdot (6 \cdot -3) \cdot (-2) \equiv (1) \cdot (1) \cdot (-2) \equiv -2 \pmod {19} $
Apr
29
comment There exists an irrational number z such that x<z<y
Consider to work with the different part in the infinite or finite decimal-digit-representations of x and y and to construct a number between them based on the first different digit...
Apr
27
comment Convergence properties of $z^{z^{z^{…}}}$ and is it “chaotic”
The Tetaration-forum database has references to 5 Baker/Rippon articles into the "zentralblatt", for instance this one: zbmath.org/?q=an:0644.30014 Perhaps from there you can proceed. The link in the Tetration-databse was eretrandre.org/rb/…. But I don't know whether this is openly available.
Apr
26
comment How many steps to reach 1? (Collatz Conjecture)
Using Pari/GP the following results occur in less than a second: . . . . . . . . . . . . . . $\small {m=3456;w=10^{m-2};\\ n1=260 *w; [n,it] = [1, 58968]\\ n1=261 *w; [n,it] = [1, 62437]\\ n1=262 *w; [n,it] = [1, 61739]\\ n1=263 *w; [n,it] = [1, 61907]\\ n1=264 *w; [n,it] = [1, 59516]\\ n1=265 *w; [n,it] = [1, 61266]\\ n1=266 *w; [n,it] = [1, 60325]\\ n1=267 *w; [n,it] = [1, 60837]\\ n1=268 *w; [n,it] = [1, 61310]\\ n1=269 *w; [n,it] = [1, 61491]\\ } $ . . . . . . . .
Apr
25
comment Regularizing the sum of all factorials
Is it sure, that Borel-summation can be applied here? I was convinced, that to be Borel-summable a series must have alternating signs (if I understood Konrad Knopp's book correctly)
Apr
25
comment Good problems to do while reading Hardy's book on divergent series?
Konrad Knopp in his book on series has a chapter about divergent series (chap 13 in the german edition) and has many exercises. (Don't have the exact english title at hand, but the book has been at least partially readable via google-books some years ago)
Apr
25
comment How many steps to reach 1? (Collatz Conjecture)
For numbers of certain structures the number of iterations can be directly determined, so for numbers of the form $n=2^m$. For numbers of the form $ n= a \cdot 2^m$ one can at least find the number of iterations down to $n' = a$ similarly. Btw., your number needs 59516 iterations to reach n'=1 ...
Apr
25
comment Existence of an $n\times n$ real matrix $A$ such that $A^2=-I$.
$A^4=I$ then $A^3 = A^{-1}$ (but I don't know whether this helps...)
Apr
25
comment Relation between Cholesky and SVD
If you can handle squareroots of negative numbers ($\to$ complex numbers) there's no problem with non-positive definite matrices.
Apr
21
comment Determining the Collatz Series as a Tree of $\forall\mathbb{N}$
@goblin: Oh sorry, "harshness" was not intended. The criterion to look at the negative numbers and ask, why there can be cycles and why not with the positive number opended my own eyes a couple of years ago when I could not make progress in the very same discussion of patterns, structures etc like in the OP's question.
Apr
16
comment Determining the Collatz Series as a Tree of $\forall\mathbb{N}$
(+1 for the effort!) However - the problem is always: can your path of proof also explain, why in the negative domain there are 3 cycles possible, although the same connectedness is existent? If you feel uneasy with negative numbers $x$ then can your proof handle the question why in the equivalent 181x+1 - problem there are two cycles possible?
Apr
5
comment Show/disprove that a system of equations expresses all the natural numbers $\forall\mathbb{N}$
I like the explorative feel (+1)
Apr
2
comment Exercising divergent summations: $\lim 1-2+4-6+9-12+16-20+\ldots-\ldots$
Dear Lubos - may I ask you to give my new answer a look and to comment about the applicability of the general ansatz?
Apr
2
comment Divergent sum of factorials
Matt, concerning your comment "OK to me..." - perhaps it is a theorem that series, which can be seen as linear compositions of finitely many formal zeta- and (Dirichlet) eta-series, can be evaluated by the same compositions of the zeta and eta-evaluations, valid also for the divergent cases (possibly except that of zeta at 1). However although I've read Hardy and Knopp I cannot remember to have seen such a theorem - perhaps is somehow implied though- I'm not an expert in formalisms. I'd be interested to see such a theorem! In principle, what I've tried in my answer, is assuming just this...
Apr
2
comment Divergent sum of factorials
(Interestingly, applying the method which I employ in my answer, the result in the linked question is $1/8$ - where the solution proposed by efferari there is off again)
Apr
2
comment Divergent sum of factorials
Ah, and look at this one. I'm not sure at the moment, whether that problem was really solved:math.stackexchange.com/questions/41263/…
Apr
2
comment Divergent sum of factorials
@Matt: look at this: go.helms-net.de/math/tetdocs/ProblemWithBellmatrix.pdf
Apr
1
comment Divergent sum of factorials
What is "wrong"? That I've tested the alternating series? That I found the results comparing two methods compatible? That the alternating-series testing should be compatible with using the eta-values instead of zeta-values?
Apr
1
comment Divergent sum of factorials
Very well! Now the last step is, that you must find an argument, that/why that reordering of summation is valid although you work with divergent series (zetas at negative arguments, or positive p in the exponent) in the third- and second-last equation. I know a very nice/instructive example where this does (counterintuitively) not work. That is the reason why I introduced the function $f_k(s)$ in my answer.