Gottfried Helms
Reputation
15,393
Next privilege 20,000 Rep.
Access 'trusted user' tools
 Nov 22 comment $2^i - 2293$ is always composite? @Peter: I just found that cookie using "Extras/Einstellungen/..." and then the register card for manual cookie-deletion.(I don't know for what reason/goal, but firefox uses one of the most uncomfortable windows I've ever seen for long lists to check/manipulate, so don't get exhausted too early) Nov 22 comment $2^i - 2293$ is always composite? @a boy, please replace your commercialized link (at adf.ly) by the direct link math.stackexchange.com/questions/597234/… . After I have followed your link I have now a "cookie" from that people on my computer which I have to find and delete manually Nov 21 comment Will values assigned to divergent series match a taylor series past the radius of convergence? As I understood things the first ($\small \ln(x-1)$ at $\small x =1$) is taken by Ramanujan's summation and the latter two by Zeta-regularization, which are very different things. The Zeta-regularization of the first series ($\sum_{k=1}^\infty k^s$ with $\small \lim_{s \to 1}$) fails and does not give a finite value to $\zeta(1)$ . Hmm, is such a statement or some equivalent one missing in the wikipedia? Nov 20 comment Fractional Euler sums? Hmm, but $\frac 12$ and $\frac 32$ are not really real numbers, which is what I read in (and expect from) your question? Nov 20 comment A limit related to super-root (tetration inverse). Hmm, just didn't think of it. Do you have a sketch of the resulting formulae applying your idea? Nov 20 comment Euler Transform elementary Proof I know only that of 1928 with the "theory of..." in the title (and is the 4'th edition of 1947). Perhaps the 1956 book is even a conceptual update but I've not yet looked inside it. Nov 20 comment A limit related to super-root (tetration inverse). @mick: sorry - what do you mean? Nov 20 answered Plotting $\left(1+\frac{1}{x^n}\right)^{x^n}$. Nov 20 comment LU Factorization of a full rank square matrix. Suggestion: it is principal (roughly "the first"/"most important") instead of principle (usually used in the term "in principle") Nov 20 answered Euler Transform elementary Proof Nov 20 comment Euler Transform elementary Proof There is a scan of K.Knopp's somehow classical book on infinite series, with chapt. 13 on divergent series and Euler-summation online available, however I only know the access to the german version (@ digicenter Goettingen) and don't know if the english translation is also accessible. Perhaps via google books; but just talking about this I know that G.H.Hardy's book is (partially) readable via that place. Nov 20 answered Standard terminology for the “quotient of a quotient” Nov 20 comment How do I evaluate this summation? I would proceed with software Pari/GP to define a function a(m,n)=if(n==0,return(1));if(m==1,return(1/n!)); .... Then I would practically compute a matrix "A" as A=matrix(16,16,r,c,a(r,c)) or the like (while I'm typing here I cannot see the formula) and after that the next formula in MW shows how to compute an actual value for th powertower of height m as sum using that coefficients of a(m,n) which are stored in the matrix plus the other sum. (I hope I didn't confuse you even more...) Nov 19 revised A limit related to super-root (tetration inverse). added 114 characters in body Nov 19 comment A limit related to super-root (tetration inverse). You can find an old version of introduction at go.helms-net.de/math/binomial_new/pmatrix.pdf . In chap 3, on page 17 I displayed my procedere. I must confess that at that time when I wrote this I was just beginning with serious work in (amateurish) number theory and the text should not yet fulfill research standards - it was merely a display and explanation of my own heuristical process. But it should give a good explanation anyway... Nov 19 comment A limit related to super-root (tetration inverse). I premultiply the vector of the original partial sums (only a finite number of terms - we'll get always only an approximation, but a "good" one), of say the first 32 or 64 terms with a certain matrix, the ESum-matrix. If the growth-rate of the original terms of the series is at most geometrix, but the signs alternate, then Euler-summation can be made arbitrarily precise by configuring the summation-matrix "ESum". Finally it boils down to have a set of coefficients for each partial sum to make a weighted sum of it. The matrix used is made from (appropriate) powers of the Pascalmatrix. Nov 18 comment Original proof of Ljunngren's equation Just trying to find some examples, I find the property where we get at least $z^2$ instead of $y^4$ (being squares) I find, that beginning with $x=7$ and the iterating $x=\text{floor} (\sqrt8 \cdot x))+3x+1$ if seem to find immediately all that $x$ which admit $z^2$ in the formula. (being $x=[7,41,239, ...]$) That only $x=7$ and $x=239$ give $z^2=y^4$ is then directly visible with numbers $x$ up to hundred digits and only some milliseconds to compute... I never related such a recursion with those type of questions. Amazing! Nov 16 revised A limit related to super-root (tetration inverse). update-notice removed Nov 15 revised A limit related to super-root (tetration inverse). added 24 characters in body Nov 15 revised A limit related to super-root (tetration inverse). added 118 characters in body