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Conjecture Reference Request

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 ...
1
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0answers
80 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 ...
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1answer
134 views

Conjectured Primality Test for Numbers of the Form k2^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
204 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
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1answer
152 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: ...
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0answers
81 views

Primality Criterion for Specific Class of Numbers of the Form kb^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
87 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
165 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
83 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
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1answer
482 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 ...
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1answer
122 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
179 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
224 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
232 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
674 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
320 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
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1answer
202 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
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1answer
194 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
415 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 ...
43
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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
287 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
741 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 $S$, ...
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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?