Use this tag if your question is about a well-known conjecture or a conjecture of your own.

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3
votes
0answers
78 views

How to prove this??

Does the following inequality hold? $p(n)\leq 2^n,$ where $p(n)$ is the $n$th prime. If this is true then it follows that: If $p(n)=p(m)^x+p(o)^y$, then $\max[x,y] \le n$.
1
vote
3answers
348 views

“Goldbach's other conjecture” and Project Euler - writing 1 as a sum of a prime and twice a square

From Problem 46 of Project Euler : It was proposed by Christian Goldbach that every odd composite number can be written as the sum of a prime and twice a square. $$9 = 7 + 2 \cdot ...
0
votes
0answers
118 views

Arithmetic progression and average of two prime numbers

Let $A=(a_n : n \in \mathbb{N})$ be the sequence given by: $$ \ a_n = a_1 + (n - 1)d,\quad a_1,\ d,\ n \in \mathbb N,\quad d\gt a_1,\quad \gcd(a_1,\ d)=1. $$ For all terms of $A$ greater than $\ ...
65
votes
23answers
1k views

Open mathematical questions for which we really, really have no idea what the answer is

There is no shortage of open problems in mathematics. While a formal proof for any of them remains elusive, with the "yes/no" questions among them mathematicians are typically not working in both ...
0
votes
0answers
42 views

How to convert an object into a sphere?

I'm not sure if I understand it enough, but doesn't the Poincare conjecture prove any shape can be turned into a sphere? How would I go about transforming such an object? Like let's say I have a ...
-1
votes
1answer
177 views

Proof for Goldbach's Conjecture [closed]

There is a proof given here. I couldn't find any flaw in it, what's wrong with it?
2
votes
1answer
311 views

Proof of Andrica when Assuming Oppermann

Proof of Andrica's conjecture by assuming Oppermann's conjecture. Oppermann's conjecture: $$n\geq2\wedge\pi\left(n^{2}-n\right) < \pi\left(n^{2}\right) < \pi\left(n^{2}+n\right).$$ ...
3
votes
1answer
159 views

Prove or disprove this upper bound on chromatic number.

Let $G$ be a simple connected finite graph and let $v \ge 4$ be the number of vertices, $E$ the number of edges, $\chi(G)$ the chromatic number , $\omega(G)$ the clique number and $\Delta$ the ...
4
votes
1answer
93 views

Measure Theory Conjecture

While I was doing some math here, I made this conjecture. Let $f_n:X\rightarrow \mathbb{R}$ be a sequence of measurable functions from the measure space $(X,\mathcal{A},\mu)$ to the measurable space ...
1
vote
1answer
83 views

Proved conjectures (now theorems) in algebra/algebraic number theory/algebraic geometry

I would like to collect some proved conjectures (not so non trivial) in algebra/algebriac number theory/algebraic geometry. For example, I consider Serre's conjecture on projective modules over ...
7
votes
3answers
343 views

On progress in mathematics: some long-open problems and long-standing conjectures

I would like to ask a question here on Math Stack Exchange taking inspiration (and therefore combining) from two well-known threads on MathOverflow: (1) Not especially famous, long-open problems which ...
18
votes
1answer
856 views

List of generally believed conjectures which cannot all be true

There are some conjectures which most leading experts believe in albeit no one can prove it yet. For example: $\mathcal{P} \neq \mathcal{NP}$, the Riemann hypothesis or the Collatz conjecture. My ...
3
votes
0answers
76 views

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

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 ...
3
votes
0answers
72 views

Big list of references [divided by categories] that collect commented open problems and conjectures [closed]

The aim of this question is to collect a big list of books or survey papers or websites which collect an up-to-date, comprehensive, well-organized, and possibly commented list of open problems. I ...
17
votes
0answers
409 views

On the problem of polynomial bijection from $\mathbb Q\times\mathbb Q$ to $\mathbb Q$

The question titled "Polynomial bijection from $\mathbb Q\times\mathbb Q$ to $\mathbb Q$" which was posed on MathOverflow attracted quite a lot of attention (and may be the question with most wrong ...
2
votes
1answer
72 views

Two conjectures regarding $\varphi(n)$

There is a famous unsolved problem called Lehmer's Totient Problem which states that, $\varphi(n)\mid n-1 \implies n$ is a prime. Where $\varphi(n)$ is Euler's Totient Function. I was ...
1
vote
1answer
91 views

Conjecture: three or more decompositions into powers with a base differing by 1 means its a perfect power

If $$(i_1)^{a_1}(i_1+1)^{b_1}=n $$ $$(i_2)^{a_2}(i_2+1)^{b_2}=n $$ $$(i_3)^{a_3}(i_3+1)^{b_3}=n $$ where all the terms are positive integers and the groups ...
6
votes
2answers
64 views

Conjecture about the powers of 2 in the prime factorizations of the series $a_n = n(n+1)(n+2)$

I found the following law and would like to know what do you think about it and if anyone can explain why this is so. Also, is this already known and proven? Consider the following series: ...
1
vote
1answer
26 views

Conjecture on the value of limit and related primality testing

Just I made a curious conjecture when I was playing with my calculator. We will use $\displaystyle\prod_{i=1}^n p_i$ where $p_i$ is the $i$-th prime. Then I have noted that, $$\left\lvert\cos ...
0
votes
0answers
16 views

Counterexample to a generalization of Gilbreath's conjecture

Consider the arrays with "initial conditions" $L_1^1>0,\ L_{n+1}^1>L_n^1,\ L_1^{i+1}=1$ satisfying the recurrence $L_n^{i+1}\in\{L_n^i-L_{\large{\inf\{m\in\Bbb Z_{>n}:L_m^i\leq ...
2
votes
0answers
72 views

A similar, but hopefully easier problem than Gilbreath's conjecture

Gilbreath's conjecture says that for every positive integer $n$, if we write out the first $n$ primes $2,3,5,7,11,13,\ldots,p_n$ take the differences between consecutive terms ...
7
votes
1answer
206 views

Zhang's theorem and Polignac's conjecture

Yitang Zhang made a groundbreaking discovery when he proved that there are infinitely many pairs of prime numbers which differ by less than $70,000,000$. Zhang's theorem has been significantly ...
13
votes
2answers
331 views

A conjecture concerning primes and algebra

A monoid morphism $\psi:\mathbb Z_+\!\!\rightarrow\mathbb Z_+$ is defined by an arbitrary function $f:\mathbb Z_+\!\!\rightarrow\mathbb Z_+$ and defines a group homomorphism $\varphi:\mathbb ...
3
votes
1answer
141 views

How to submit a conjecture for review?

I was recently attempting to solve one of the more known, already proved, problems in mathematics when I stumbled across an observation I thought might be worth digging into further. Unfortunately I ...
1
vote
0answers
62 views

Inequality with Euler's totient function

In A conjecture concerning primes and algebra on MSE, I defined a multiplicative function $\omega:\mathbb Z_+\!\!\to\mathbb Z_+$ with $\omega(p_n)=n$, for the $n$-th prime $p_n$. It was conjectured ...
8
votes
3answers
328 views

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

Question: Which inequalities similar to the famous isoperimetric inequality is known? conjectured? I recently learned about some inequalities which are all similar to the famous isoperimetric ...
45
votes
1answer
2k views

Conjecture $\int_0^1\frac{\mathrm dx}{\sqrt{1-x}\ \sqrt[4]x\ \sqrt[4]{2-x\,\sqrt3}}\stackrel?=\frac{2\,\sqrt2}{3\,\sqrt[8]3}\pi$

$$\int_0^1\frac{\mathrm dx}{\sqrt{1-x}\ \sqrt[4]x\ \sqrt[4]{2-x\,\sqrt3}}\stackrel?=\frac{2\,\sqrt2}{3\,\sqrt[8]3}\pi\tag1$$ The equality numerically holds up to at least $10^4$ decimal digits. ...
12
votes
5answers
1k views

What does proving the Collatz Conjecture entail?

From the get go: i'm not trying to prove the Collatz Conjecture where hundreds of smarter people have failed. I'm just curious. I'm wondering where one would have to start in proving the Collatz ...
8
votes
2answers
270 views

Why Goldbach's conjecture is difficult to prove?

Why Goldbach's conjecture is still non-solved and is difficult to prove? What makes the mathematicians fail when trying to prove it?
6
votes
1answer
175 views

making mathematical conjectures

If a non-mathematician wanted to conjecture something and had strong numerical evidence to support the conjecture, how would he/she go about doing so? Would the mathematical community (a) take it ...
5
votes
1answer
106 views

At most n functions

Some background: I was trying to solve the functional equation f(f(x))=sin(x). I realized that $f(\pi n)$ is a root of f for all integers n, because $f(f(\pi n))=\sin(\pi n)=0$. Thus, we can write f ...
4
votes
4answers
90 views

Can decimals/fractions be odd or even? [duplicate]

At school I asked the question and I kept wondering "Can fractions or decimals be odd or even?"
18
votes
1answer
701 views

Examples of falsified (or currently open) longstanding conjectures leading to large bodies of incorrect results.

In general, the way that modern mathematical research is conducted isn't the way that many would assume is the ideal method of research. That is, mathematics is not the linear progression of ...
13
votes
1answer
571 views

Big-Daddy-Conjectures and Hierarchy of Mathematical Conjectures

I am interested in the Hierarchy and Connections between various different open problems in Mathematics, and the most general conjectures in various fields of Mathematics. Examples of Hierachy ...
2
votes
1answer
45 views

Firoozbakht's conjecture and maximal gaps

In the Wikipedia article, it seems to me as if it's implied that it is enough to check the conjecture only for maximal gaps (numbers $n$ s.t. $\forall k<n:g_n>g_k$). I.e it holds that ...
0
votes
2answers
79 views

Check my proof of Lehmers conjecture [closed]

$\phi{(n)}=n-1$ for $n$ being composite. Here, $\phi{(n)}$ represents the Euler totient function. (1-1/p1)(1-1/p2)......(1-1/pn)=((n-1)/n) because this will prove that Phi of n=(n-1).. We need to ...
64
votes
12answers
7k views

Conjectures that have been disproved with extremely large counterexamples?

I just came back from my Number Theory course, and during the lecture there was mention of the Collatz Conjecture. I'm sure that everyone here is familiar with it; it describes an operation on a ...
55
votes
2answers
1k views

Integral $\int_1^\infty\frac{\operatorname{arccot}\left(1+\frac{2\pi}{\operatorname{arcoth}x-\operatorname{arccsc}x}\right)}{\sqrt{x^2-1}}\mathrm dx$

Consider the following integral: $$\mathcal{I}=\int_1^\infty\frac{\operatorname{arccot}\left(1+\frac{2\,\pi}{\operatorname{arcoth}x\,-\,\operatorname{arccsc}x}\right)}{\sqrt{x^2-1}}\mathrm dx,$$ where ...
-1
votes
1answer
76 views

A Conjecture on The Generalization of Quadratic Reciprocity Law

Is there any way to prove the following conjecture regarding the Generalization of Quadratic Reciprocity Law. The statement being, $$ \left(\dfrac{a_1}{a_2}\right)\left(\dfrac{a_2}{a_3}\right) ...
1
vote
2answers
399 views

twin prime conjecture

Whether I am correct or wrong I don't know. If there are any corrections, please let me know. Let $p_n$ = product of all primes. (of course we can go still beyond as we know $p_n$ is infinite). Now ...
11
votes
2answers
125 views

Trigamma identity $4\,\psi_1\!\left(\frac15\right)+\psi_1\!\left(\frac25\right)-\psi_1\!\left(\frac1{10}\right)=\frac{4\pi^2}{\phi\,\sqrt5}.$

I heuristically discovered the following identity for the trigamma function, that I could not find in any tables or papers or infer from existing formulae (e.g. [1], [2], [3], [4], [5], [6]): ...
4
votes
2answers
159 views

What if a conjecture were provably unprovable?

Suppose we found a proof that "The Twin Prime Conjecture cannot be proven", without any conclusion as to the conjecture itself being true or false. Is it then possible for the conjecture to be true? ...
0
votes
0answers
40 views

Which conjectures imply Standard conjectures?

My attention (around this question) arose from this "MO Question" in which Jakob says: The period and the Hodge conjecture imply the Standard conjectures. Is that true? Are there other ...
45
votes
1answer
2k views

Does $|n^2 \cos n|$ diverge to $+\infty$?

I was recently exposed to the problem of deciding whether $$ \lim_{n \to +\infty} |n \cos n| = +\infty$$ where the limit is taken over the integers. As $|\cos n|$ oscillates throughout the interval ...
3
votes
0answers
99 views

conjecture about prime numbers and distance between them

is there a name for this conjecture? Conjecture: given $p_n$ a prime number sequence where $p_1=2,p_2=3,\cdots$, then for all $n\in\mathbb{N}^*$ and $k\in\mathbb{N}$, holds that $\displaystyle ...
5
votes
2answers
740 views

Erdös-Straus conjecture

I'm reading a lot about the Erdös-Straus Conjecture (ESC), a conjecture that states that for every natural number $p \geq 2$, there exists a set of natural numbers $a, b, c$ , such that the following ...
6
votes
1answer
127 views

Mathematical conjectures believed to be false

I just came about the Firoozbakht's conjecture, and read that it is believed to be wrong, as it would contradict some heuristic methods. However, the conjecture is numerically verified for ...
3
votes
3answers
788 views

Goldbach conjecture

I have seen the following in my observation. I am so sorry, if I am wrong. I do not know how far I am right. But, I feel that, it might be correct. If correct, Please let me know that way to proceed ...
0
votes
2answers
981 views

Checking the Harald Helfgott proof of the little Goldbach conjecture without a public release of numerical checks?

A few month ago, a proof of the little/ternary Goldbach conjecture has been claimed by Harald Helfgott with three articles: Major arcs for Goldbach's theorem Minor arcs for Goldbach's theorem ...
23
votes
1answer
434 views

Simplify $\frac{_3F_2\left(\frac{1}{2},\frac{3}{4},\frac{5}{4};1,\frac{3}{2};\frac{3}{4}\right)}{\Pi\left(\frac{1}{4}\big|\frac{1}{\sqrt{3}}\right)}$

Is it possible to simplify the ratio $$\mathcal{E}=\frac{_3F_2\left(\frac{1}{2},\frac{3}{4},\frac{5}{4};\ 1,\frac{3}{2};\ \frac{3}{4}\right)}{\Pi\left(\frac{1}{4}\Big|\frac{1}{\sqrt{3}}\right)},$$ ...