Can $\gcd(n^k \pm 1, \hspace{2mm} n! \pm 1)>1$ have arbitrarily many but finitely many solutions?

Fix an integer $k \geq 2$ and let $n \geq 2$ be integer as well. Let $\lambda_1,\lambda_2 \in \{-1, 1 \}$, and then consider the set $$ S_{k}^{\lambda_1,\lambda_2} := \{n \in \mathbb{N}\setminus\{0,1\} : \gcd(n^k+\lambda_1, \hspace{2mm} n!+\lambda_2) > 1\}. $$

Previous discussions here case $k=2, \lambda_1=\lambda_2=-1$, case $k=3, \lambda_1=\lambda_2=-1$ (thanks to Wilson's theorem) have shown that these sets exhibit very different behaviors depending on $k$ and the choice of signs: e.g., we have proved that the choice $(k=2, \lambda_1=\lambda_2=-1)$ implies that $S_k^{\lambda_1,\lambda_2} = \varnothing$, whereas $(k=3, \lambda_1=\lambda_2=+1)$ implies that $S_k^{\lambda_1,\lambda_2}$ has the same cardinality of the prime numbers set (i.e., $\aleph_0$).
Now, in several cases, numerical evidence suggests that $S_k^{\lambda_1,\lambda_2}$ is nonempty but we have only found a small number of solutions (sometimes isolating only a single pair of solutions).

Question. Let $\#S_k^{\lambda_1,\lambda_2}$ denote the number of values of $n$ satisfying $\gcd(n^k+\lambda_1, \hspace{2mm} n!+\lambda_2) > 1$ for the given choice of $k$, $\lambda_1$, and $\lambda_2$ (e.g., $\#S_2^{-1,-1} = 0$, $\#S_3^{1,1} = +\infty$, etc.). Is it true that, for every integer $c \geq 1$ there exists an integer $k \geq 2$ (and a choice of signs $\lambda_1,\lambda_2$) such that $c < \# S_k^{\lambda_1,\lambda_2} < +\infty$?

In other words, can one obtain arbitrarily large but finite sets of solutions, or does one always fall into one of the following regimes: no solutions, finitely many (bounded), or infinitely many?

Remark. It is worth noting that the $\gcd$ can also be non-squarefree(!). For example, I have found that the choice $k=10$, $\lambda_1=\lambda_2=-1$ is enough to prove this fact, as $S_{10}^{-1,-1} = \{9, 15, 29, 52, \ldots \}$, and so the element $n=9$ implies that $\gcd(9^{10}-1,\hspace{2mm} 9!-1)=121=11^2$ (a perfect square).

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Are concatenations of two consecutive Mersenne numbers which are congruent to 6 mod 7 necessarily composite?

In this question on MSE, Enzo Creti asks for a prime number formed by concatenating the Mersenne numbers $2^n-1$ and $2^{n-1}-1$, for example, $40952047$. For all residues modulo $7$, he found primes except for the residue $6$. This is somewhat surprising because the residue $1$ occurs only with half frequency.

Is there any hidden structure forcing a non-trivial factor in the case of residue $6$, or was it just "bad luck" that no prime was found despite an enormous search range?

I invite everyone to join in the search for a prime. I posted the necessary details on github.

The following vector contains all numbers $n\le 366800$ leading to a prime:

$[2, 3, 4, 7, 8, 12, 19, 22, 36, 46, 51, 67, 79, 215, 359, 394, 451, 1323, 2131, 3336, 3371, 6231, 19179, 39699, 51456, 56238, 69660, 75894, 79798, 92020, 174968, 176006, 181015, 285019, 331259, 360787, 366770]$

Exponent $541456$ leads to another probable prime with residue $5 \bmod 7$ and $325990$ digits, but it need not be the next in increasing order. More details can be found on the github-site. Heuristically, for every $k\ge 3$, the range $[10^k\ldots 10^{k+1}]$ should contain $5.4$ numbers leading to a prime, so the range $[10^4\ldots10^5]$ with $8$ primes is "above average", whereas the range $[10^3\ldots 10^4]$ is within the expectation. The sequences of exponents and associated primes are here: A301806 and here: A298613. I noticed these things: let's call $$\DeclareMathOperator{\ec}{ec} \ec(n)=2^n-1\Vert2^{n-1}-1 $$ where $\Vert$ denotes the concatenation in base $10$. Then

• $\ec(43\cdot5)$ is prime. $5$ (odd) is congruent to $-8 \pmod {13}$.
• $\ec(43\cdot 1620)$ is prime. $1620$ (even) is congruent to $8 \pmod{13}$.
• $\ec(43\cdot 2140)$ is prime. $2140$ (even) is congruent to $8 \pmod{13}$.
• $\ec(43\cdot 12592)$ is prime. $12592$ (even) is congruent to $8 \pmod{13}$.

Moreover $43\cdot5=41\cdot5+10$, $43\cdot1620=41\cdot1699+1$, $43\cdot2140=41\cdot2244+16$, $43\cdot12592=41\cdot13206+10$. This implies that $$ \frac{43\cdot5}{41}, \frac{43\cdot1620}{41}, \frac{43\cdot2140}{41}, \frac{43\cdot12592}{41} $$ will have a repeating term $\overline{02439}$ or in other words that they are of the form $41s+r$, where $r$ is an integer in the set $(1,10,16,18,37)$

Remark: there are two numbers in the sequence $1323$ and $39699$ which have the form: $$ \left(44\cdot 10^n+\frac{(10^n-1)}{9}\right)\cdot 3^n. $$ Cheating and using Gemini I found this polynomial $-14521k^2 + 282802k - 176261$.

For k=1,2,3 It gives Three terms of the sequence:

$92020$ $331259$ and $541456$

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Mathematical games interesting to both you and a 5+-year-old child

Background: My daughter is 6 years old now, once I wanted to think on some math (about some Young diagrams), but she wanted to play with me... How to make both of us to do what they want ? I guess for everybody who has children, that question comes up. Okay, I said to her: let's play a game which I called "Young diagram" for her: we took a sheet of paper and I tried to explain to her what a Young diagram is, she was asked to draw all the diagrams of some size n=1,2,3,4,5...

Question: Do you have some experience/proposals of "games" which you can play with your children, which would be on the one hand would make some fun for them, on the other would somehow develop their logical/thinking/mathematical skills, and on the other hand would be of at least some interest for adult mathematicians ?

Related MO questions:

“Mathematics talk” for five year olds it is quite related to the present question, but slightly different - it is about a single presentation to children, while the present question is about your own children with whom you play everyday, you can slightly "push", and so on...

How do you approach your child's math education? it is also related, but the present questions has a slightly different focus: games interesting for children and adults. The book by Alexandre Zvonkine, "Math for little ones" (in Russian here), recommended in answer there - is really something related to the present question.

Which popular games are the most mathematical? is NOT directly related, but may serve as kind of inspiration for answers...

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I think Allen Knutson's answer on “Mathematics talk” for five year olds:

I've spoken (to 5+ years old) about the "puzzles" that Terry Tao and I developed for Schubert calculus, like the left two here:

can be a nice example of an answer to the present question as well: on the one hand there is something to explain to the child and some colorful pictures, and on the other hand that is about research level math ...

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Global regularity for 3D Navier-Stokes with arbitrary large H¹ initial data - Claim for Clay Millennium Problem [closed]

Claim: I present a proof of global regularity for the 3D incompressible Navier-Stokes equations with arbitrary large H¹ initial data.

Main result: For divergence-free $u_0 \in H^1(\mathbb{R}^3)$, the unique Leray-Hopf weak solution satisfies $$ \|u(t)\|_{L^2}^2 \leq C (1+t)^{-\delta} $$ for some $\delta > 0$, with $C$ depending only on $\|u_0\|_{H^1}$. This energy decay rules out finite-time blowup and establishes global regularity.

Key idea: Viscous self-cancellation via negative feedback. The nonlinear term generates fine scales that enhance dissipation, yielding the closed differential inequality $$ \frac{d}{dt}\|u\|_{L^2}^2 + \frac{\nu}{2}\|\nabla u\|_{L^2}^2 \leq 0 $$ for arbitrary large data, without small-data assumptions.

Full paper: https://doi.org/10.5281/zenodo.20689937

Part I (foundational energy decay): https://doi.org/10.5281/zenodo.20656332

I am seeking expert feedback specifically on:

1. The closure of the feedback inequality for arbitrary $H^1$ data
2. The viscous self-cancellation mechanism in Section 3

All comments welcome.

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What is a good approximating space for the following space of pairs of functions?

Let $\Omega^+$ be a finite polygonal domain in $\mathbb{P}^1(\mathbb{C})$, by which we mean that its boundary is a polygon with only finite points.

Let $$ \Omega^- = \mathbb{P}^1(\mathbb{C}) \setminus \overline{\Omega^+}.$$

Let $(f, g)$ be a pair of functions, where $f$, resp. $g$, is a continuous complex-valued function on $\overline{\Omega^+}$, resp. $\overline{\Omega^-}$, which is holomorphic on $\Omega^+$, resp. $\Omega^-$. Denote by $S$ the space of such pairs of functions.

I am interested in approximating $S$ by a finite-dimensional space, for the sake of numerical calculations. It is reasonable to approximate $f$ and $g$ by complex polynomials, but I believe such approximations do not behave well near the boundary of $\Omega^+$.

Define the function $h$ to be

$$ h = \operatorname{res}(f) - \operatorname{res}(g),$$

where $h$ is a continuous complex-valued function on the boundary of $\Omega^+$ and where $\operatorname{res}$ denotes the restriction to the boundary of $\Omega^+$.

It would seem that $h$ would have to be approximated too (perhaps by a finite Laurent polynomial?). This reminds me of hyperfunctions, assuming I got the name right, which I am not familiar with.

Motivation: finding a numerical approximation of the solution to a problem in magnetostatics involving a polygonal magnetic material.

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Regarding the Geometric Langlands conjecture for positive characteristic

My doubt is regarding the following paper of Gaitsgory, Raskin

https://arxiv.org/abs/2508.02237.

The target of the above paper was to construct the following functor

$$\mathbb{L}^{\mathrm{restr}}_{\check G} : \mathrm{Shv}_{\mathrm{Nilp}}(\mathrm{Bun}_G) \rightarrow \mathrm{IndCoh}_{\mathrm{Nilp}}(\mathrm{LS}^{\mathrm{restr}}_{\check G})$$

However, what they ended up achieving on the rhs was not $\mathrm{IndCoh}_{\mathrm{Nilp}}(\mathrm{LS}^{\mathrm{restr}}_{\check G})$ but $\mathrm{IndCoh}_{\mathrm{Nilp}}(\mathrm{'LS}^{\mathrm{restr}}_{\check G})$ where $\mathrm{'LS}^{\mathrm{restr}}_{\check G}$ is the disjoint union of some (may not be all) of connected components of $\mathrm{LS}^{\mathrm{restr}}_{\check G}$. Only in the specific case of $G=GL_n$, the above functor is achieved. This is the content of Theorem 1.3.9(iii) in the paper.

My question is which part of the proof does not generalize to an arbitrary reductive algebraic group. As per my understanding we would need something in the line of Frenkel-Gaitsgory-Vilonen work for $GL_n$, which has been used in the proof of Theorem 1.3.9(iii), that needs a generalization to arbitrary $G$. Is it the only part that do not get generalized to an arbitrary group or there is something else? For example in the following paper, Scholze explicitly mentions the issue which I did not fully understand.

https://people.mpim-bonn.mpg.de/scholze/Exp1252_Scholze.pdf

In Pg- 18 Scholze say " Currently, the argument does not fully work because of subtleties with the theory of singular support for étale sheaves in mixed characteristic — this is ultimately responsible for the appearance of the open and closed subset $\mathrm{'LS}^{\mathrm{restr}}_{\check G}\subset\mathrm{LS}^{\mathrm{restr}}_{\check G}$".

What are the subtleties that Scholze is referring to? Thanks, in advance.

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Discrete condensed objects

Let $\mathrm{Prof}^{\mathrm{light}}$ be the site of light profinite sets. For a presentable $\infty$-category $\mathcal{C}$, let $\mathrm{Cond}(\mathcal{C})$ denote the $\infty$-category of hypercomplete $\mathcal{C}$-valued sheaves on $\mathrm{Prof}^{\mathrm{light}}$. Then the functor $\mathrm{Cond}(\mathcal{C})\to \mathcal{C};\; F\mapsto F(\ast)$ admits a left adjoint functor $\mathcal{C}\to \mathrm{Cond}(\mathcal{C});\;X\mapsto \underline{X}$. My questions are:

(1) Is the functor $\mathcal{C}\to \mathrm{Cond}(\mathcal{C});\;X\mapsto \underline{X}$ fully faithful?

(2) Is there a characterization of the essential image of the above functor?

At least for (1), I think it should be a formal consequence of the fact that $\ast\in \mathrm{Prof}^{\mathrm{light}}$ is a projective object, but I couldn't find a reference. Any comments or references would be welcome.

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Question/References on the Skorokhod M1 topology

Let $D(0,T)$ be the space of right continuous functions with left limits defined on $[0,T]$. Consider the Skorokhod M1 topology on $D(0,T)$, see e.g. S. Ledger, Skorokhod’s M1 topology for distribution-valued processes (ProjectEuclid link DOI) for its definition. Denote by $\rho$ the corresponding metric. I have the questions as follows :

1. Is the space $\bigl(D(0,T),\rho\bigr)$ separable?

2. Let $(\mathbb P_n)_{n\ge 1}$ be a weakly convergent sequence of probability measures on $D(0,T)$ endowed with $\rho$. Denote by $\mathbb P$ its limit. Is there a dense subset $I\subset [0,T]$ s.t.

$$\mathbb P_n\circ \phi_t^{-1} \xrightarrow{n\to\infty} \mathbb P\circ \phi_t^{-1},\quad \quad \quad \text{for all } t\in I,$$

where $\mathbb P_n\circ \phi_t^{-1}$ denotes the image measure of $\mathbb P_n$ under $\phi_t$ and $\phi_t:D(0,T)\to\mathbb R$ is defined by $\phi_t(f):=f(t)$ for any $f\in D(0,T)$?

I am unable to find the related references. Any answers or references are highly appreciated.

PS: The Skorokhod M1 topology is not the Skorokhod J1 topology that is used mostly. This topology M1 is strictly weaker than J1, and is defined by the so called parametric representation of graphs.

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When are products of definable sets definable?

Say that a class of structures $\mathbb{K}$ (all of the same similarity type) is product-nice iff, whenever $\mathcal{M}_1,...,\mathcal{M}_n\in\mathbb{K}$ and $\varphi(\overline{x})$ is a formula in the language of $\mathbb{K}$, the relation $\prod \varphi^{\mathcal{M}_i}$ is parameter-freely-definable in the product structure $\prod\mathcal{M}_i$. (It's important to use the same formula on each "factor," since otherwise we could trivialize things by permuting the factor structures.) Similarly, say that a theory $T$ is product-nice iff its model class $\mathit{Mod}(T)$ is product-nice, and that a structure $\mathcal{M}$ is product-nice iff $\{\mathcal{M}\}$ is product-nice.

Most structures (so a fortiori most theories and classes in general) are not product-nice. For example, $(\mathbb{R};+,0)$ is not product-nice since the set $\{(a,b):a\not=0\not=b\}$ is not even fixed by automorphisms in the group $\mathbb{R}^2$. On the other hand, the theory of cancellative semirings is product-nice, essentially because we can "locate the axes" - think about the nonzero zero-divisors $\alpha$ such that no nonzero zero-divisor $\beta$ has $\{\gamma: \beta\gamma=0\}\subsetneq\{\gamma: \alpha\gamma=0\}$. (Meanwhile, two axis elements are on the same axis iff their sum is again an axis element or is zero.)

I'm going to teach a logic class in the fall, and one thing I want to make sure I give due respect to is the Feferman-Vaught theorem and the situation(s) with regard to its converse. (Generally, I want students to appreciate the nuances of direct products more than I did way back when!) So my question is:

What are some criteria, ideally of a first-order theory but more generally of an individual structure, which ensure or forbid product-niceness?

I'm especially interested in the case where the theory in question is equational (i.e. which varieties are product-nice? which algebras generate product-nice varieties?), since this would help contrast the behaviors of equational and first-order logic, which I've noticed has been a stumbling block for students in the past.

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How far can the geometric median be from the mean under variance normalization?

Let $X$ be a random variable in $\mathbb{R}^d$ with law $\mu$. We denote by $\mathcal{P}(\mathbb{R}^d)$ the set of all Borel probability measures on $\mathbb{R}^d$.

Assume that $\mu \in \mathcal{P}(\mathbb{R}^d)$ has finite second moment. Its mean is $$ \bar{\mu} = \mathbb{E}_\mu[X], $$ and its variance is $$ \sigma^2(\mu) = \mathbb{E}_\mu\big[\|X - \mathbb{E}_\mu[X]\|^2\big]. $$

Define the geometric median of $\mu$ as any minimizer of $$ m \mapsto \mathbb{E}_\mu\big[\|X - m\|\big]. $$ We denote it by $\mathrm{GeoMed}(\mu)$. It is known that the minimizer is unique if $\mu$ is not supported on a line.

We consider the quantity $$ \Phi(\mu) = \frac{\|\mathrm{GeoMed}(\mu) - \bar{\mu}\|^2}{\sigma^2(\mu)}. $$

Question. Is the quantity $$ \sup_{\substack{\mu \in \mathcal{P}(\mathbb{R}^d) \\ \sigma^2(\mu) > 0}} \Phi(\mu) $$ finite?

If yes:

• can one characterize (or approximate) extremal distributions?
• does the supremum depend on the dimension $d$, or is it dimension-free?

Motivation: while both the mean and the geometric median capture a notion of consensus, they behave differently in the presence of outliers. This question is motivated in part by adversarial machine learning, where the geometric median has robustness properties that the mean does not.

Any references on quantitative bounds comparing the geometric median and the mean under only a second-moment assumption would be very helpful.

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Is this non-commutativity of temporal morphisms a known categorical construction? [closed]

I'm working on a framework for financial state representation and have arrived at a categorical structure I'd like to understand better. I'm not a category theorist by training, so I'd welcome corrections to the framing.

The construction:

Let C be a category where:

• Objects are economic states (the state of an obligation at a given moment)
• Morphisms are three primitive operators: R̂ (Recognition), Â (Articulation), Ŝ (Settlement)
• A complete transaction is the composition: T̂ = Ŝ ∘ Â ∘ R̂

The non-commutativity observation:

The relative ordering of R̂ and Ŝ determines the classification of the resulting object, not whether the composition is valid. Specifically:

• R̂ before Ŝ → the unsettled interval produces one type of object (a receivable: value recognized, not yet received)
• Ŝ before R̂ → the unsettled interval produces a different type of object (a deferred obligation: value received, not yet earned)

Both orderings are valid compositions; they land in different objects in the target. This has been empirically validated across a working prototype.

This suggests [R̂, Ŝ] ≠ 0, but the content of the non-commutativity isn't "one order is forbidden" — it's "the two orderings map to different objects." I'm not certain what the right categorical language is here. Is this standard non-commutativity in the category-theoretic sense, or is it better described as the two compositions landing in different components of a coproduct, or something else entirely?

The two-party question:

Any exchange involves two parties, each with their own R̂/Â/Ŝ timeline for the same underlying event. The empirically validated state space has exactly 16 states (2 directions × 2³ timing combinations).

I've conjectured that this 16-state space might be expressible as the set of natural transformations between two functors — one for each party's perspective mapping the same underlying event into their respective internal state categories. If so, the 16-state bound would be categorically necessary rather than just empirically observed.

Questions:

1. Is the non-commutativity framing above standard, or is there a more precise categorical description of "two valid orderings landing in different objects"?

2. Is the natural transformation hypothesis for the 16-state space a sensible construction, and if so, does it add anything beyond the empirical observation?

---

Context: this arises from work on financial state representation (Dukketta Research Initiative, papers forthcoming on SSRN). Happy to provide more detail on the empirical validation if useful.*

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If $(\alpha_t)$ is $\mathbb{F}^X$-progressive for a continuous process $(X_t)$, can we write $\alpha_t = \tilde{\alpha}(t,X)$?

Let $X = (X_t)_{t \geq 0}$ be a continuous, real-valued process defined on some probability space $(\Omega,\mathcal{F},P)$, and let $\mathbb{F}^X = (\mathcal{F}_{t}^X)_{t \geq 0}$ be the filtration generated by $X$. Suppose that $\alpha = (\alpha_t)_{t \geq 0}$ is real-valued and $\mathbb{F}^X$-progressive. Can we write

• $\alpha_t(\omega) = \tilde{\alpha}(t,X(\omega))$ for some product measurable function $\tilde{\alpha} : \mathbb{R}_+ \times C(\mathbb{R}_+;\mathbb{R}) \rightarrow \mathbb{R}$

or

• $\alpha_t(\omega) = \tilde{\alpha}(t,X_{t \land \cdot}(\omega))$ for some product measurable function $\tilde{\alpha} : \mathbb{R}_+ \times C(\mathbb{R}_+;\mathbb{R}) \rightarrow \mathbb{R}$ ?

Here $C(\mathbb{R}_+;\mathbb{R})$ denotes the space of continuous functions from $\mathbb{R}_+$ to $\mathbb{R}$.

The answer does not seem to be a simple application of the Doob-Dynkin factorization lemma + functional monotone class argument as the progressive $\sigma$-algebra (from my understanding) is not generated by a random variable nor does it have "nice" elementary generators.

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A truncated Newton expansion for an explicit floor sequence coming from zeta sums [closed]

Question. For each positive integer $k$, define the finite polynomial

$$P_k(x)=(-1)^{k+3}\sum_{i\geq0}\left(\left\lfloor \frac{2(2(4k+i))^{k+1}-1}{3}\right\rfloor-\left\lfloor \frac{2(2(4k+i))^{k+1}}{3}-\frac{2(2(4k+i))^k(4(4k+i)+9)}{27\cdot 2^{2(4k+i)}}\right\rfloor\right)\sum_{j=0}^{i}(-1)^j\binom{k+2}{j}x^{i-j}.$$

The sum is finite because the difference of floors vanishes for all sufficiently large $i$. Is it true that, for every positive integer $k$ and every integer $w\geq0$,

$$\begin{aligned} &\sum_{j=k+2}^{w}\binom{w}{j}\sum_{m=0}^j(-1)^m\binom{j}{m}\left\lfloor \frac{2(2(4k+j-m))^{k+1}}{3}-\frac{2(2(4k+j-m))^k(4(4k+j-m)+9)}{27\cdot 2^{2(4k+j-m)}}\right\rfloor\ &=[x^w]\left(\frac{(-1)^{\left\lfloor\frac{k-1}{2}\right\rfloor}3^{\left\lfloor k/2\right\rfloor}x^{k+2}(1+x)^{1-\operatorname{Mod}(k,2)}}{(1-x)^{k+2}(1+x+x^2)}+\frac{x^{k+2}}{(1-x)^{k+2}}P_k(x)\right)? \end{aligned}$$

Computationally, this identity has been checked exactly for $1\leq k\leq100$ and $1\leq w\leq200$, with larger spot checks also passing. The correction polynomial $P_k(x)$ is zero for $1\leq k\leq27$. The first correction occurs at $k=28$, where $P_{28}(x)=-1$.

Motivation.

Let $D^0(x_i)=x_i$ and $D^{j+1}(x_i)=D^j(x_i)-D^j(x_{i-1})$. Then

$$D^j(x_i)=\sum_{m=0}^j(-1)^m\binom{j}{m}x_{i-m}.$$

Newton's backward expansion gives

$$x_i=\sum_{j=0}^i\binom{i}{j}D^j(x_j).$$

Substituting the formula for $D^j(x_j)$ gives

$$x_i=\sum_{j=0}^i\binom{i}{j}\sum_{m=0}^j(-1)^m\binom{j}{m}x_{j-m}.$$

Now fix a positive integer $k$ and apply this to the even subsequence of a function $f$ by setting

$$x_i=f(8k+2i)=f(2(4k+i)).$$

Then

$$f(8k+2w)=\sum_{j=0}^w\binom{w}{j}\sum_{m=0}^j(-1)^m\binom{j}{m}f(2(4k+j-m)).$$

I then asked what happens if this Newton expansion is truncated at $j=k+1$. Define

$$F_k(N)=\left\lfloor \frac{N^k\sum_{i=2}^{N}\zeta(i)}{\sum_{i=2}^{N}(-1)^i\zeta(i)}\right\rfloor.$$

The truncated residual is

$$R_k(w)=F_k(8k+2w)-\sum_{j=0}^{k+1}\binom{w}{j}\sum_{m=0}^j(-1)^m\binom{j}{m}F_k(2(4k+j-m)).$$

Using the full Newton identity, this is equivalently the Newton tail

$$R_k(w)=\sum_{j=k+2}^{w}\binom{w}{j}\sum_{m=0}^j(-1)^m\binom{j}{m}F_k(2(4k+j-m)).$$

Computationally, for even $N\geq8k$, the zeta-ratio floor appears to simplify to

$$F_k(N)=\left\lfloor \frac{2N^{k+1}}{3}-\frac{2N^k(2N+9)}{27\cdot 2^N}\right\rfloor.$$

Thus, replacing $N$ by $2(4k+j-m)$, the Newton tail becomes the explicit floor tail

$$\sum_{j=k+2}^{w}\binom{w}{j}\sum_{m=0}^j(-1)^m\binom{j}{m}\left\lfloor \frac{2(2(4k+j-m))^{k+1}}{3}-\frac{2(2(4k+j-m))^k(4(4k+j-m)+9)}{27\cdot 2^{2(4k+j-m)}}\right\rfloor.$$

For $1\leq k\leq27$, the correction polynomial $P_k(x)$ vanishes, so the conjecture reduces to the simpler formula

$$R_k(w)=[x^w]\frac{(-1)^{\left\lfloor\frac{k-1}{2}\right\rfloor}3^{\left\lfloor k/2\right\rfloor}x^{k+2}(1+x)^{1-\operatorname{Mod}(k,2)}}{(1-x)^{k+2}(1+x+x^2)}.$$

For example, when $k=5$, this gives

$$R_5(w)=[x^w]\frac{9x^7}{(1-x)^7(1+x+x^2)},$$

whose coefficients begin

$$0,0,0,0,0,0,9,54,189,513,1188,2457,4671,8316,\ldots.$$

When $k=8$, this gives

$$R_8(w)=[x^w]\left(-\frac{81x^{10}(1+x)}{(1-x)^{10}(1+x+x^2)}\right).$$

Starting at $k=28$, the small exponential term inside the floor becomes large enough to change the floor. This is why the correction polynomial $P_k(x)$ is needed. For $k=28$, one gets $P_{28}(x)=-1$.

The first few nonzero correction polynomials are

$$P_{28}(x)=-1,$$

$$P_{29}(x)=2,$$

$$P_{30}(x)=59-2x,$$

$$P_{31}(x)=410-29x+x^2,$$

$$P_{32}(x)=4110-437x+30x^2-x^3.$$

The rational generating function also has a sparse factorial expansion. The basic identity is

$$[x^w]\frac{Cx^a}{(1-x)^a(1+x+x^2)}=C\sum_{i\geq0}(-27)^i\left(\binom{w}{a+6i}-2\binom{w}{a+1+6i}+3\binom{w}{a+2+6i}-3\binom{w}{a+3+6i}+9\binom{w}{a+5+6i}\right),$$

with the convention that $\binom{w}{s}=0$ for $s>w$. In the present identity, one takes $a=k+2$ and $C=(-1)^{\left\lfloor\frac{k-1}{2}\right\rfloor}3^{\left\lfloor k/2\right\rfloor}$; when $k$ is even, the extra factor $1+x$ adds the same expansion shifted by one.

The sparse factorial expansion comes from the algebraic identity

$$\frac{1-2y+3y^2-3y^3+9y^5}{1+27y^6}=\frac{1+y}{1+3y+3y^2}.$$

Indeed, since

$$\sum_{w\geq0}\binom{w}{s}x^w=\frac{x^s}{(1-x)^{s+1}},$$

the sparse factorial ladder has the kernel

$$\frac{1-2y+3y^2-3y^3+9y^5}{1+27y^6},\qquad y=\frac{x}{1-x}.$$

The algebraic identity turns this kernel into

$$\frac{1+y}{1+3y+3y^2}.$$

Since $y=x/(1-x)$, this becomes

$$\frac{1-x}{1+x+x^2},$$

explaining the denominator $1+x+x^2$ in the base term.

The part I do not understand is why this explicit floor sequence produces this rational generating function plus the finite correction polynomial $P_k(x)$.

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Interpretation of $a_p(f)$ for the abelian variety attached to a weight 2 newform $f$

We know that there is a correspondence between modular forms $f \in S_2^{\mathrm{new}}(N)$ and abelian varieties of $\mathrm{GL}_2$-type over $\mathbb{Q}$. For $f \in S_2^{\mathrm{new}}(N)$, what arithmetic data does the $p$th coefficient $a_p(f)$ provide about the associated abelian variety $A_f$ (or its reduction mod $p$, to be more precise)? I believe we have that the trace of $p$-Frobenius of $A_f$ is $T_p(f):=\operatorname{Tr}_{K_f/\mathbb{Q}}(a_p(f)) =\sum_{\sigma}\sigma(a_p(f)),$ where $\sigma : K_f \hookrightarrow \mathbb{C}$ are the complex embeddings of the Hecke field $K_f$, but I'm more interested in the algebraic number $a_p(f)$ itself.

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When is a (commutative) $k[\varepsilon]$-algebra isomorphic to a subalgebra of $M_n(k[\varepsilon])$ for some $n$?

Let $k$ be a field. Is it possible to characterize when a commutative $k[\varepsilon]$-algebra (where $k[\varepsilon] :\cong k[x]/x^2$) which is finite-dimensional over $k$ embeds into $M_n(k[\varepsilon])$ as a $k[\varepsilon]$-subalgebra for some $n$? It's necessary that $\operatorname{ann}(\varepsilon)^2=0$. Is it sufficient?

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On convex polygons that have multiple least area circumscribed triangles that are mutually non congruent

Question: For a convex polygon P with N vertices, let f(P) denote the number of pairwise non-congruent triangles that simultaneously achieve the minimum area among all circumscribed triangles of P. What is max f(P) over all convex N-gons? Is this maximum unbounded as N → ∞, or is there a universal bound independent of N?

To my knowledge, existing work is on finding min area circumscribed triangles efficiently and not enumerating them with the non congruence constraint.

A natural extension would be to construct convex polygons that have a specified number of mutually non congruent circumscribed triangles. An explicit construction for f(P)= 2 or 3 would be nice.

Note 1: The question applies to general convex regions (including smooth ones) in addition to polygons.

Note 2: A natural variant question emerges if we replace area by perimeter.

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Reference request: reflection–translation commutators versus local curvature holonomy

I am looking for a reference, or a standard name, for the following obstruction. I am not claiming novelty; I am trying to find the right existing language.

There are two elementary "defect" phenomena that look formally similar.

1. On the line R\mathbb{R} R, let ρ(x)=−x\rho(x)=-x ρ(x)=−x be a reflection and τs(x)=x+s\tau_s(x)=x+s τs​(x)=x+s a translation. Then ρτsρ−1=τ−s\rho\tau_s\rho^{-1}=\tau_{-s} ρτs​ρ−1=τ−s​, so with the convention [g,h]=ghg−1h−1[g,h]=ghg^{-1}h^{-1} [g,h]=ghg−1h−1,

[ρ,τs]=ρτsρ−1τs−1=τ−2s,[\rho,\tau_s]=\rho\tau_s\rho^{-1}\tau_s^{-1}=\tau_{-2s},[ρ,τs​]=ρτs​ρ−1τs−1​=τ−2s​, a translation by twice the offset. Likewise on S1S^1 S1, a reflection and a rotation give [ρ,Rθ]=R−2θ[\rho,R_\theta]=R_{-2\theta} [ρ,Rθ​]=R−2θ​ (up to sign convention and mod 2π2\pi 2π). Note the leftover defect element here is itself a translation/rotation — orientation-preserving; the reflection is one of the two generators, not the result.

2. For a bundle with connection, parallel transport around a small loop gives a holonomy element; infinitesimally the curvature is the commutator of covariant derivatives,

R(X,Y)=[∇X,∇Y]−∇[X,Y],R(X,Y)=[\nabla_X,\nabla_Y]-\nabla_{[X,Y]},R(X,Y)=[∇X​,∇Y​]−∇[X,Y]​, i.e. holonomy around a small contractible loop is governed, to leading order in the enclosed area, by curvature, in the Ambrose–Singer sense.

Read loosely, both fit one slogan: a closed comparison of two non-commuting operations leaves a defect element.

What I am not asking about. I know the affine / Riemann–Cartan picture genuinely unifies translational and rotational transport defects — torsion as translational holonomy, curvature as rotational holonomy, both in an affine/Cartan framework (Kobayashi–Nomizu vol. I; Katanaev's geometric theory of defects; Hehl–Obukhov). I also know transport functors give a categorical account of parallel transport on the bundle side (Schreiber–Waldorf). So I am not asking whether torsion and curvature sit in one structure — they do.

The narrower question. Is there a standard framework in which the reflection–translation commutator of case 1 is treated as an instance of the same mechanism as local curvature holonomy in case 2? Every attempt I make runs into what look like two obstructions:

In the line/circle example the non-commutation comes from an orientation-reversing involution conjugating a shift to its inverse, ρτρ−1=τ−1\rho\tau\rho^{-1}=\tau^{-1} ρτρ−1=τ−1. The base is flat; the mechanism is algebraic conjugation by a reflection. If one tries to read the two generators as local connection transports, the reflection generator is not naturally available: the restricted holonomy group (generated by local curvature data, for a connected base) is connected and so sits in the identity component of the structure group, whereas ρ\rho ρ is orientation-reversing. (A reflection can appear as global monodromy on a non-orientable or disconnected setting, but that looks like a flat/topological mechanism, not local curvature holonomy.) On a 11 1-dimensional base there is no curvature 22 2-form at all, since Λ2T∗M=0\Lambda^2 T^*M = 0 Λ2T∗M=0; local curvature holonomy in the usual sense needs a two-dimensional infinitesimal loop. So the line example cannot literally be a curvature effect — the defect τ−2s\tau_{-2s} τ−2s​ comes from the reversal, not from curvature.

So the two phenomena appear to share only the form "commutator = defect," while the mechanism differs (flat orientation-reversal vs. curvature), and the natural rigorous home for case 2 does not accommodate the reflection of case 1.

Questions.

Is this distinction — flat orientation-reversal commutator versus local curvature holonomy — a standard, named distinction or obstruction? Is there a reference that states it cleanly? Conversely, is there a known framework — for example groupoid holonomy, Cartan geometry with a disconnected structure group, reflection/Coxeter groups, orbifolds, or flat bundles with monodromy — in which the reflection–translation commutator really is a recognizable instance of the same statement as local curvature holonomy, rather than only sharing the word "commutator"? (I mention these only as possible places to look; an answer needn't address all of them.)

I would be grateful for references in either direction: a bridge I have missed, or a clean account of why this is not usually treated as a common connection-holonomy mechanism. Thank you.

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Cohomological interpretation of gluing conditions

This question comes from a problem of theoretical physics. Stated in its simplest form, there is a complex line bundle over $S^1$. For each $z \in S^1$, the fiber $F_z$ is the complex eigenspace of a unimodular real 2x2 matrix $M$ (it is understood that $|tr(M)|<2$), the other eigenspace is the fiber over $1/z$. As such, the line bundle is trivial, but there is a further structure: there is a $\mathbb{Z}/2$ action $F_z \to F_{1/z}$ such that $U \in F_z \to \sigma U \in F_{1/z}$ where $\sigma=\begin{pmatrix}1 & 0 \\ 0 & -1\end{pmatrix}$.

What I am looking for is an equivariant section $z\to U(z)$, that is, satisfying the gluing condition $U(z)=\sigma U(1/z)$. Using a cover of $S^1$ with 2 open sets $o_1$ and $o_2$ overlapping at $z=1$ and $z=-1$, it is easy to see that it suffices to consider the gluing condition at these two points only. Now, while this can always be done at one of the two points $z=\pm 1$, it cannot always be done at the other point $1/z$. It may happen that the gluing condition be $U^1(1)=-\sigma U^2(1)$.

So I have a $\mathbb{Z}/2$ action on $S^1$ represented by $z \to 1/z$ and a $\mathbb{Z}/2$ action on $F$ represented by $\sigma$, but I could also choose $-\sigma$. So somehow, I have 2 group actions on the fibers and I would like to interpret the fact that both have to be used as a non-triviality of the bundle, i.e., the fact to add a "$-1$" as an obstruction to having a continuous equivariant section.

I would like to have a description of this situation in terms of a cohomological space. From the gluing conditions, this looks like Cech cohomology with "twisted" coefficients, but I do not know how to put his in clear mathematical form.

Any insight would be greatly appreciated.

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Finding a finite covering for a Julia set

In his paper "A general theory of self-similarity", Tom Leinster develops a theory that expresses facts about self-similar objects taking advantage of coalgebras, universal properties and, in general, categorical language.

I am trying to apply some of their results to a particular kind of self-similar objects, the so-called Newton's fractals: given a complex function $f(z)$, the Newton-Raphson method for finding roots of the function gives a map $N_f(z) := z- \frac{f(z)}{f'(z)}$ (often referred to as Newton iteration map). The Julia set $J(N_f)$ of this map is almost always a self-similar space.

The particular function I am working on is $$f(z)=z^3-1,$$ which gives raise to a $J(N_f)$ like the one represented in white in this picture from Wikipedia:

Julia set for the function

In order to apply a theorem from the paper to this Julia set, I need to define explicit finite coverings of the set.

The point is: I do not really know what this set is. Surely my knowledge of complex dynamics, fractal geometry and so on is weak. I come from the more categorical side of this discussion. But I am not really finding anything, in discussions, in books etc, about how to get an explicit, geometric, expression of a specific Julia set. I know the definition, I know some properties, I can plot pictures, but nothing more.

Actually, the picture I included in this question gives a clear idea of what this $J(N_f)$ is, in my opinion. On the boundary of the three main basins I put a sequence of drop-like shapes. On the boundary of these drops I put other drop-like shapes, and so on. I believe that $J(N_f)$ is the "limit" of this process. But I cannot be sure, can I?

How could I prove it? How can I prove that a family of sets I defined actually covers that Julia set? Do explicit descriptions of Julia sets (something of the kind: $\{ z \in \mathbb{C} \; \text{such that} \dots \} $) exist?

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Semidirect product representations of finitely presented groups

I've been looking into the $(m,n,p;q)$ groups, defined as $$G=\langle a,b\mid a^m,b^n,(ab)^p,[a,b]^q\rangle.$$ Generally, as you increase the parameters, the groups seem to fall into three categories:

1. Finite groups
2. "Tame" infinite groups, which seem to have some discernable structure
3. "Wild" infinite groups, those which have infinitely many finite simple quotients, or have other complications

I'm interested in the second category, the groups that are infinite but have some structure. Specifically, I would like to try to classify (at least some of) those groups as a (potentially iterated) semidirect product of a free abelian group with a finite group.

Here are some examples of what I mean:

$(2,2)$: $\langle a,b\mid a^2,b^2\rangle$ is isomorphic to the infinite dihedral group, which is the semidirect product of $\Bbb Z$ (generated by $ab$) and $\Bbb Z_2$ (generated by $a$).

$(2,3,6)$: $\langle a,b\mid a^2,b^3,(ab)^6\rangle$ is isomorphic to $\Bbb Z^2\rtimes\Bbb Z_6$ ($\Bbb Z^2$ generated by $ab^2ab,abab^2$ and $\Bbb Z_6$ generated by $ab$)

$(2,4,4)$ is isomorphic to $\Bbb Z^2\rtimes\Bbb Z_4$

$(3,3,3)$ is isomorphic to $\Bbb Z^2\rtimes\Bbb Z_3$

$(2,6,6;2)$ took a lot more work... I had to iterate Reidemeister-Schreier algorithm three times. An index $4$ subgroup is the group $\langle x,y,z\mid x^2,y^3,z^3,(xy)^3,(xz)^3,(yz)^2\rangle$, for which every pair of the three generators generates $A_4$. An index $3$ subgroup of that was a free product of two Klein-$4$-groups quotiented by one strange relator. Finally, the free abelian group of rank $3$ was an index $4$ subgroup of that. After some more work I was able to get to the representation of $(2,6,6;2)$ as $((\Bbb Z^3\rtimes A_4)\rtimes\Bbb Z_2)\rtimes\Bbb Z_2$ (The original step to an index $4$ subgroup had to be split into two steps of taking an index $2$ subgroup). Can this be simplified?

I'm sure someone else has tried to do something similar with these groups. Is there an article someone can recommend? What ranks of free abelian groups are encountered?

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