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

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11
votes
0answers
199 views
+50

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
63 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
87 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
58 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
24 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
14 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
67 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
194 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
314 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 ...
62
votes
22answers
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 ...
3
votes
1answer
132 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
57 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 ...
7
votes
3answers
302 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 ...
44
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
254 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
171 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 ...
3
votes
4answers
77 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?"
16
votes
1answer
563 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
557 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
41 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
75 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 ...
62
votes
12answers
6k 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 ...
54
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
69 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
398 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
121 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
147 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
35 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 ...
44
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
94 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
4answers
695 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 ...
5
votes
1answer
111 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 ...
0
votes
0answers
113 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 $\ ...
3
votes
3answers
770 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
940 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
430 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 $$\mathcal{E}=\frac{_3F_2\left(\frac{1}{2},\frac{3}{4},\frac{5}{4};\ 1,\frac{3}{2};\ \frac{3}{4}\right)}{\Pi\left(\frac{1}{4}\Big|\frac{1}{\sqrt{3}}\right)},$$ ...
2
votes
2answers
80 views

Low Level Books on Conjectures/Famous Problems

I am currently an undergraduate math/CS major with coursework done in Linear Algebra, Vector Calculus (that covered a significant amount of Real 1 material), Discrete Math, and about to take courses ...
1
vote
2answers
69 views

Intuitively, what separates Mersenne primes from Fermat primes?

A Mersenne prime is a prime of the form $2^n-1$. A Fermat prime is a prime of the form $2^n+1$. Despite the two being superficially very similar, it is conjectured that there are infinitely many ...
3
votes
1answer
28 views

Find all $n$ so that $c_n$ $>$ $\pi(n^2)$

Find all $n$ $\in$ $\mathbb{N}$ so that $p_{c_n}$ $>$ $n^2$ where $p_n$ denotes the $n$-th prime and $c_n$, the $n$-th composite. I have tried doing the problem using The stronger version of ...
2
votes
0answers
62 views

Is this a conjecture or an already existing one??

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
0answers
27 views

Non-repeating decimals in 1/n [duplicate]

Here's a conjecture I made: The number of non repeating decimal places in the base-ten representation of the fraction 1/n, where n is an integer, is equal to whichever is higher: the exponent of 2 in ...
2
votes
0answers
30 views

Decidability of $P = NP$?

(Please, don't sign this as duplicate of this question, they are not.) Is it possible, that the well-known $P=NP$ conjecture is undecidable in ZFC? Is there any result about this topic?
2
votes
2answers
106 views

Equal perimeters of squares and right angled isosceles triangles

Consider a square ABCD having length l and breadth. Now start folding the sides AB and AC so that the figure becomes something like this $$$$ All the vertical and horizontal folds/stairs are equal in ...
5
votes
2answers
2k views

Which notable mathematicans have tried solving the Riemann hypothesis?

I have read that the Riemann hypothesis is the most important open question in mathematics and has been open since 1859. I am wondering which famous mathematicians have actually tried to solve it and ...
0
votes
1answer
91 views

It's sufficient to prove collatz conjecture for $3+6k, k \geq 0$?

Thinking about this problem, I saw two interesting properties of Collatz graph. Firstly, if we consider that every even number $e$ can be represented (on a single way) as $e = o 2^n$, where $o$ is an ...
2
votes
0answers
63 views

Prove the inequality for composite numbers

Is it true that $c_m+c_n$ $>$ $c_{m+n}$ for all $m$, $n$ $\in$ $\mathbb{N}$? Though the result seems true, I can't get a solution. Even the bounds on $c_n$ obtained from Prime Number Theorem ...
5
votes
1answer
326 views

Prove that there exists an $m$ such that for any $n>m$ there exists at least one prime between $c_n$ and $n$

Let $c_n$ be the $n$-th composite. Then the problem is to prove that- $\pi(c_n)-\pi(n)>0$ $\forall n>m$ I have tried to progress in the problem using an elementary approach. So far I have ...
1
vote
1answer
163 views

Tangent at average of two roots of cubic with one real and two complex roots

I was able to easily prove that the tangent at the average of two roots of a real cubic polynomial passed through the third root of the function. But I have only done this for functions with three ...