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0answers
42 views

Conjectured new primality test for Mersenne numbers

How to prove that this conjecture about a new primality test for Mersenne numbers is true ? Definition: Let $M_{q}=2^{q}-1 , S_{0} = 3^{2} + 1/3^{2} , \ and: \ S_{i+1} = S_{i}^{2}-2 \pmod{M_{q}}$ ...
-4
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0answers
87 views

Why should “small” P be preferred? [on hold]

In contrast, of course, is the approach of finding an NP language of super-polynomial complexity. But why the overwhelming, obvious yet implicit favoritism? Has it anything to do with our ...
5
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1answer
168 views

Connes Embedding Conjecture and Fusion Categories

I was recently introduced to Connes' Embedding Conjecture (CEC) which states: Every separable type $II_{1}$ factor is embeddable into $R^{\omega}$. Where $\omega$ is a generic free ultrafilter on ...
0
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1answer
155 views

A conjecture on the prime counting function

I was thinking about the Second Hardy-Littlewood conjecture for quite sometime (some of my posts are related to this). In one of my earlier post I conjectured that the inequality ($\pi(x)$ denotes the ...
39
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2answers
2k views

Arctangents of odd powers of the golden ratio

While trying to answer this MSE question, I found that arctangents of many odd powers of the golden ratio $\varphi=\frac{1+\sqrt5}2$ are expressible as rational linear combinations of arctangents of ...
0
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1answer
170 views

Is the following conjecture equivalent to the Second Hardy-Littlewood Conjecture? [closed]

Let $y$ be an arbitrary positive real number such that $y\ge 2$. Then if we can prove that, $$\lim_{x\to\infty}\dfrac{\pi(x)+\pi(y)}{\pi(x+y)}=1$$will it imply that for all sufficiently large $x$ ...
13
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1answer
536 views

What are the strongest conjectured uniform versions of Serre's Open Image Theorem?

This question concerns the uniform conjectured effective versions and generalizations of these two results of Serre on $\ell$-adic Galois representations $\rho_{E,\ell}$ associated to a non-CM ...
5
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0answers
216 views

What will be the consequences if second Hardy-Littlewood conjecture turns out to be true?

It is generally believed that the Second Hardy-Littlewood Conjecture is false. But it has not been proved (or disproved) yet. My question is, What would be the consequences if Second ...
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3answers
2k views

The resolution of which conjecture/problem would advance Mathematics the most? [closed]

This is an impossibly broad question, and makes the unwarranted assumption that Mathematics is a uniform field. It might make more sense to ask the same question restricted to, say, Mathematical ...
3
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0answers
572 views

Second Hardy-Littlewood Conjecture theme

If Second Hardy-Littlewood Conjecture is true then we can claim that $\pi(x)-\pi(y)\leq \pi(x-y)$. Thus the conjecture gives an upper bound for the number of primes between $x$ and $y$. I have found ...
2
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0answers
92 views

Non-negative, monotone polynomial sequences without combinatorial interpretation

I am wondering what sequences of integers there are, that are known to grow polynomially, are non-negative, monotone but lacks a combinatorial interpretation. By combinatorial interpretation, they ...
0
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1answer
194 views

Conjectured Primality Test for Numbers of the Form $k \cdot 2^n+1$ with $n>2$ [closed]

Definition : Let $P_m(x)=2^{-m}\cdot \left(\left(x-\sqrt{x^2-4}\right)^m+\left(x+\sqrt{x^2-4}\right)^m\right) $ where $m$ and $x$ are positive integers . Conjecture : Let $N=k\cdot 2^n+1$ with ...
6
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1answer
227 views

Is the Tate conjecture known for etale covers of products of curves

Let $X$ be a (smooth projective geometrically connected) surface over a finitely generated field $k$. The Tate conjecture predicts that, for $l$ a prime number invertible in $k$, the Chern class map ...
3
votes
1answer
175 views

If there are two primes at an even distance $k$ is there another disjoint pair of primes also at distance $k$?

Suppose $p,q$ are two primes at even distance $k$. Must there necessarily exist a different pair $p',q'$ composed of entirely different numbers such that $p'$ and $q'$ are also at distance $k$? Edit: ...
2
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0answers
114 views

Primality Criterion for Specific Class of Numbers of the Form $k\cdot b^n-1$

Let $N=k\cdot b^n-1$ where $b$ is an even integer , $3\nmid b$ , $3\nmid N$ , $k \equiv 1,5 \pmod{6}$ , $k< b^n $ and $n>2$ . Let $S_i=P_b(S_{i-1})$ with $S_0=P_{k\cdot b/2}(P_{b/2}(4))$ , ...
2
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0answers
104 views

Primality Criterion for Specific Classes of Generalized Fermat Numbers

Let $F_n(b)= b^{2^n}+1 $ where $b$ is an even integer , $ 3\nmid b , 5\nmid b $ and $n\ge2$ Let $S_i=P_b(S_{i-1})$ with $S_0=P_{b/2}(P_{b/2}(8))$ where ...
8
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0answers
198 views

Consequences of Zeeman's conjecture

Recall the Zeeman's conjecture: if $K$ is a contractible polyhedron of dimension 2, then $K\times I$ has a collapsible subdivision. Zeeman showed that this implies the Poincaré conjecture in ...
2
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0answers
119 views

Isoperimetric inequality, isodiametric inequality, hyperplane conjecture… what are the inequalities of this kind known or conjectured?

I duplicate here a question I asked on math.stackexchange. Question: Which inequalities similar to the famous isoperimetric inequality is known? conjectured? I recently learned about some ...
2
votes
1answer
614 views

Need help publishing a mathematical proof? [closed]

Im not a mathematician with profession, but I know a group working on a Beal's Conjecture for years. They think that they have found a proof but the problem is that they don't know how to publish ...
2
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1answer
141 views

Conjectures in classical harmonic function theory

Because I'm doing research in the area of harmonic function theory I would like to know are there any conjectures in the theory of harmonic functions in $\mathbb{R}^{n}$ still open. I know that there ...
1
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1answer
193 views

Representing the integers with powers of 2 and 3

Suppose that I have a number of the form $$ x = \frac{1}{3^m}(2^{h} - \sum\limits_{k=1}^{m}3^{m-k}2^{v_k} ) $$ where m is a positive integer, and $v_1 = 0 $ $v_{k+1} = v_k +1$ for $1 \leq k ...
5
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1answer
279 views

Who is attributed with the conjecture that every multiply-perfect number greater than $1$ is even?

I know that Descartes is considered to be the first to ask whether or not odd perfect numbers exist ($n$ such that $\sigma(n)=2n$, where $\sigma(n)$ is the sum of divisors of $n$), and he also ...
7
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1answer
278 views

Vaught conjecture for uncountable languages

Recall Vaught conjecture: the number of countable models of a first-order complete theory in a countable language is finite or $\aleph_0$ or $2^{\aleph_0}.$ Now let $\lambda$ be an uncountable ...
11
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2answers
770 views

On Generalizations of Fermat's Conjecture

We know the following facts: (1) For all $1\leq n\leq 2$ the equation $x_{1}^{n}+x_{2}^{n}=x_{3}^{n}$ has a solution in $\mathbb{N}$. (2) For all $3\leq n$ the equation ...
6
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0answers
357 views

Graphs with graphic imbalance sequences

Let $G$ be simple undirected graph and $e=uv\in E(G)$. The imbalance of the edge $e$ is the value $imb(e)=|d(u)-d(v)|$. Let $M_{G}$ denotes the imbalance sequence (or more correctly, multiset of ...
9
votes
1answer
280 views

An analogue of the Bass-Quillen conjecture with power or Laurent series

The famous Quillen-Suslin theorem (formerly known as Serre's problem/conjecture) states that every projective module over $k[x_1,\dots, x_n]$ is free for $k$ a field. Replacing $k$ by a more general ...
11
votes
1answer
234 views

An elementary expression for $_3F_2(1,1,9/4;2,2;-1)$

Consider the following series: $$S=\sum_{n=1}^\infty\frac{(-1)^n\ \Gamma\left(\frac{5}{4}+n\right)}{n^2\ \Gamma(n)}.$$ It can be expressed in terms of a hypergeometric function: ...
9
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1answer
419 views

A conjecture on intersection of some intervals.

It was proved here that if $a\in \mathbb{N}_{\geq3}$ then $$\bigcap_{i = 1}^{a} \bigcup_{j = 0}^{i-1} \left[\frac{1+aj}{i},\frac{a(j+1)-1}{i}\right] = \varnothing \tag{1}$$ It may be conjectured ...
45
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7answers
2k views

How closed-form conjectures are made?

Recently I posted a conjecture at Math.SE: $$\int_0^\infty\ln\frac{J_\mu(x)^2+Y_\mu(x)^2}{J_\nu(x)^2+Y_\nu(x)^2}\mathrm dx\stackrel{?}{=}\frac{\pi}{2}(\mu^2-\nu^2),$$ where $J_\mu(x)$ and $Y_\mu(x)$ ...
12
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2answers
305 views

A sequence based on Catalan–Mihăilescu problem

It was conjectured by Catalan in 1844 that the only solutions of the equation $x^a-y^b=1$ over variables $a,b,x,y\in\mathbb{N^+}$ are trivial ones: $3^1-2^1=1$ and $3^2-2^3=1$. The conjecture was ...
13
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1answer
791 views

Status of Grothendieck's conjecture on homomorphisms of abelian schemes

In [1] Grothendieck posits the following: Conjecture. Let $S$ be a reduced connected scheme, locally of finite type over Spec($\mathbf{Z}$) or a field $k$, $A$ and $B$ two abelian schemes over $S$, ...
5
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4answers
1k views

What are conjectures that are true for primes but then turned out to be false for some composite number?

Note: This is an update formulation since many people misunderstood the question before. Of course it is easy to make a statement like "Every n is a prime or at most 1000", which is true for every ...
11
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5answers
1k views

Is there a progress on a solution of the inequality $\pi (m+n) \leq \pi (m) + \pi (n)$

in 1923 Hardy and Littlewood proposed the conjecture $\pi (m+n) \leq \pi (m) + \pi (n)$. Is there any progress towards solving this conjecture?