@TedShifrin I wrote an answer firstly defining a "connectable number" and then changed it to "connectible number". Do you think they are ok? Is there a better name for suggesting that a sequence can be connected in another one?
@Ted Yes, and I wasn't suggesting that you were suggesting otherwise. Just trying to volunteer as a good person to ask. Of course, that doesn't preclude the possibility that my ramblings on this forum may be error-prone.
For example, let's call an operation good-sequence-behaved-three-term-wise if for all a, b, c we get a* (b*c) = (a*b) * c. That's not good -- just say "associative".
@Ted No, I don't but alarm bells should be raised when someone inexperienced in maths wants to coin a new term. It's a bit analogous to a software developer wanting to write their own square-root function instead of using a standard implementation.
@TedShifrin this cames from this question, I wanted to say that I can find a "nice sequence" (such that the sum of adjacent numbers is always prime) for two numbers and "connect" them to form a third one. For it to happen, I need the last number ($n$) to be either in the start or in the end of the sequence (like a LEGO piece)
Someone might just have the talent or knack for coming up with better, more efficient, or more descriptive notations. We all have unique talents, so we shouldn't discount ourselves @Ted
@TedShifrin sure. Consider a nice sequence (we can always find one for a prime twin by defining a sequence $S_p$, wait a bit) of length, say, $11$. I can construct it as $(1,10,3,8,5,6,7,4,9,2,11)$ (note the pattern) and take another one of length $k\leq(p+1)/2 $, say 4. The four one can be simply $(1,2,3,4)$ (connectible, the $4$ must be in the end). Now I construct another one of length $p-k+1=11-4+1=8$ by the following algorithm
@Ian: I looked at the link. So I see only one sequence there. I'm still not sure what you intend. Given sequence $\{a_n\}$ you want a new sequence $\{b_m\}$ with the property that $b_m = a_{f(m)}$ for some function $f\colon\Bbb N\to\Bbb N$?
This discrete math stuff is just so not my way of thinking .... AGH.
@TedShifrin I want a sequence such that the sum $a_k +a_{k+1}$ is always prime, right? Ok. I can always make it for a prime twin $p$ in $(p, p+2)$. Just take $(a_n)_1^p$ with $a_{2k-1}=2k-1$ and $a_{2k}=p+1-2k$
@TedShifrin A connectible sequence is a sequence of length $n$ such that the sum of any adjacent numbers is prime and $n$ is at the end or at the start of the sequence. A number $n$ is connectible if there is a connectible sequence of length $n$.
Because I'm no longer doing research and I contribute by doing the best job I can of teaching. The sad thing is that so many of the students don't reciprocate by trying to do their best. If I were going to quit caring and doing my best, I would already have retired.
@TedShifrin ok, let's redefine some things. A finite sequence $S$ is interesting if it is a permutation of $\{1,2,\ldots, n\}$ and it either starts or ends with $n$. A number $n$ is interesting if there is an interesting sequence $S$ with $|S|=n$. Is that definition ok? I know $S$ is not a set, but $|S|$ seems to fit nicely
I'm ok with it, @Ian, but I would skip the first step. Just define it for the natural number $n$. $n$ is interesting if I can re-order the set $1,2,\dots,n$ so that ...
I still talk math with colleagues and help them with their research, @Balarka, and same with grad students, but my heart has always been in teaching undergraduates, and I've concentrated on that for the last ten years or so.
@TedShifrin I see. I would be much more interested in researching than teaching peoples, if I ever get the chance. By the way, are you a frog or a bird?(ams.org/notices/200902/rtx090200212p.pdf)
@TedShifrin Ok, now the claim is that given a prime pair $(p, p+2)$, $p$ and $p+1$ are always interesting. Moreover, if $2k-1\leq p$ and $k$ is interesting, then $p+1-k$ is interesting. That is the main theorem in the answer
Are anyone here familiar with Roy's approach to Schanuel's conjecture? I just finished Zilber fields, and am trying to move onto Roy conjecture and differential operator theory.
Is it me or am I wandering through many things at once?
@TedShifrin You could just set your preferred tags on the front page to, say, only DiffGeo or similar and ignore the mass of homework questions. Or just show up to chat and nowhere else like me :P
@Chri'ssis I conjecture that the minimal extension in which $\psi(1+x)+\psi(1-x)$ can be expressed contains $\psi(x)$.
On the other hand, the same is not true for $\text{Li}_2(1+x)+\text{Li}_2(1-x)$, as it can be expressed in $\mathcal{EL}$ for $x = i$, namely, $\frac{-1}{2}\log(-i)^2$
Interesting, isn't it?
@cyril Yes, then it's not always known. What you refer to is known to us transcendental number theorists as 'linear independence'
Probably the best result known to us is that at least one of $\pi + e$ and $e^{\pi^2}$ is transcendental.
(The former can be extended to algebraic independece)
Interesting. I found myself working with something titled "An Example By Littlewood" which seemingly started with ring theory and ends up with presentation theory. Not only that, but an interesting reference is given 'I. Davis and Balarka Sen. Hardy-Ramanujan, stochastic systems and maximality methods. Journal of Singular Lie Theory'
The coolest thing is that researchers actually submitted some of these abstruse papers for publication in online math journals which granted publications!!! Although these journals are freely available for reading, the publication is quite expensive...
Question for set-theorists in here, is there any trick to prove that if <A,<> is a well ordered subset of <B,<>, and both A and B are infinite, that <A,<> is alike to <B,<>?
Oh. I think I can say that there is a one-to-one function, since if B is infinite yet countable, |B|=|A|.. so I just have to prove that it also keeps the order..
It is, and both are infinite so |B|=|A| and I get that there is one-to-one function.. question is how do I prove that it keeps the order.. they do have the same order, so it should be easy. And yet I am stuck.
@Charlie I'm a bit concerned I'm far too quick-tempered, irritable in the last period of time. I should get some rest, 4, 5 hours / night don't seem enough. High temperature has similar effects on me, but this time there is a different cause. :-)
Hey all, I have embarrassing question. You know in division, when you divide, for example, 8 by 3, you get 2 with a remainder of something. If you drop the remainder, you just have 2. What is that called?
the fact that 3 goes into 8 two times, does that have a name?
Good evening, someone knows how can I prove : that given infinitely many points, there is always other one which has an irrational distance from all of them. Please
