Questions tagged [conjectures]

For questions related to conjectures which are suspected to be true due to preliminary supporting evidence, but for which no proof or disproof has yet been found

Filter by
Sorted by
Tagged with
224
votes
19answers
36k 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 ...
12
votes
4answers
3k views

Erdős-Straus conjecture

The Erdős-Straus Conjecture (ESC), states that for every natural number $n \geq 2$, there exists a set of natural numbers $a, b, c$ , such that the following equation is satisfied: $$\frac{4}{n}=\...
24
votes
2answers
6k views

The $5n+1$ Problem

The Collatz Conjecture is a famous conjecture in mathematics that has lasted for over 70 years. It goes as follows: Define $f(n)$ to be as a function on the natural numbers by: $f(n) = n/2$ if $n$ ...
48
votes
3answers
1k views

Any odd number is of form $a+b$ where $a^2+b^2$ is prime

This conjecture is tested for all odd natural numbers less than $10^8$: If $n>1$ is an odd natural number, then there are natural numbers $a,b$ such that $n=a+b$ and $a^2+b^2\in\mathbb P$. $\...
35
votes
4answers
4k views

Proving a known zero of the Riemann Zeta has real part exactly 1/2

Much effort has been expended on a famous unsolved problem about the Riemann Zeta function $\zeta(s)$. Not surprisingly, it's called the Riemann hypothesis, which asserts: $$ \zeta(s) = 0 \...
16
votes
2answers
545 views

How to show an infinite number of algebraic numbers $\alpha$ and $\beta$ for $_2F_1\left(\frac13,\frac13;\frac56;-\alpha\right)=\beta\,$?

(Note: This is the case $a=\frac13$ of ${_2F_1\left(a ,a ;a +\tfrac12;-u\right)}=2^{a}\frac{\Gamma\big(a+\tfrac12\big)}{\sqrt\pi\,\Gamma(a)}\int_0^\infty\frac{dx}{(1+2u+\cosh x)^a}.\,$ There is also $...
47
votes
3answers
2k views

A conjectural closed form for $\sum\limits_{n=0}^\infty\frac{n!\,(2n)!}{(3n+2)!}$

Let $$S=\sum\limits_{n=0}^\infty\frac{n!\,(2n)!}{(3n+2)!},\tag1$$ its numeric value is approximately $S \approx 0.517977853388534047...$${}^{[more\ digits]}$ $S$ can be represented in terms of the ...
18
votes
3answers
654 views

Conjecture $\Re\,\operatorname{Li}_2\left(\frac12+\frac i6\right)=\frac{7\pi^2}{48}-\frac13\arctan^22-\frac16\arctan^23-\frac18\ln^2(\tfrac{18}5)$

I numerically discovered the following conjecture: $$\Re\,\operatorname{Li}_2\left(\frac12+\frac i6\right)\stackrel{\color{gray}?}=\frac{7\pi^2}{48}-\frac{\arctan^22}3-\frac{\arctan^23}6-\frac18\ln^2\!...
16
votes
3answers
696 views

conjectured general continued fraction for the quotient of gamma functions

Given complex numbers $a=x+iy$, $b=m+in$ and a gamma function $\Gamma(z)$ with $x\gt0$ and $m\gt0$, it is conjectured that the following general continued fraction which is symmetric on $a$ and $b$ is ...
12
votes
0answers
170 views

Infinitely $more$ algebraic numbers $\gamma$ and $\delta$ for $_2F_1\left(a,b;\tfrac12;\gamma\right)=\delta$?

Given the complete elliptic integral of the first kind $K(k_\color{blue}m)$, Dedekind eta $\eta(\tau)$, j-function $j(\tau)$, and hypergeometric $_2F_1\left(a,b;c;z\right)$ with $\color{brown}{a+b=c=\...
12
votes
2answers
414 views

A conjectured value for $\operatorname{Re} \operatorname{Li}_4 (1 + i)$

In evaluating the integral given here it would seem that: $$\operatorname{Re} \operatorname{Li}_4 (1 + i) \stackrel{?}{=} -\frac{5}{16} \operatorname{Li}_4 \left (\frac{1}{2} \right ) + \frac{97}{...
6
votes
1answer
635 views

Is it known or new? [duplicate]

Possible Duplicate: Starting digits of 2^n While I was randomly working with number patterns, I came along with some interesting pattern which seems to turn to a conjecture in fact. My ...
0
votes
1answer
377 views

Subsubgroups are subgroups of subgroups / Multiplicative Property of the Index

Algebra by Michael Artin Prop 2.8.14 Multiplicative Property of the Index Statement of Prop 2.8.14 Let $G \supseteq H \supseteq K$ be subgroups of a group G. Then $[G:K] = [G:H][H:K]$. Proof of ...
82
votes
2answers
4k views

Conjecture $\int_0^1\frac{dx}{\sqrt[3]x\,\sqrt[6]{1-x}\,\sqrt{1-x\left(\sqrt{6}\sqrt{12+7\sqrt3}-3\sqrt3-6\right)^2}}=\frac\pi9(3+\sqrt2\sqrt[4]{27})$

Let $$\alpha=\sqrt{6}\ \sqrt{12+7\,\sqrt3}-3\,\sqrt3-6\,=\,\big(2+\sqrt{3}\big) \big(\sqrt{2} \sqrt[4]{27}-3\big)\,=\,\frac{3\sqrt{3}}{3+\sqrt2\ \sqrt[4]{27}}.\tag1$$ Note that $\alpha$ is the unique ...
22
votes
2answers
885 views

Euler's Totient function $\forall n\ge3$, if $(\frac{\varphi(n)}{2}+1)\ \mid\ n\ $ then $\frac{\varphi(n)}{2}+1$ is prime

While I was studying Euler's Totient function, $\varphi(n)$, I stumbled upon the marvelous book "Index to Mathematical Problems, 1980-1984" By Stanley Rabinowitz. In this page of the book (...
24
votes
1answer
856 views

Status of a conjecture about powers of 2

I recently saw a conjecture on a blog ( http://blog.tanyakhovanova.com/?p=311 ) which the author refers to as the 86 conjecture. The conjecture claims that all powers of 2 greater than $2^{86}$ have a ...
8
votes
2answers
256 views

the ratio of jacobi theta functions and a new conjectured q-continued fraction

Given the squared nome $q=e^{2i\pi\tau}$ with $|q|\lt1$, define $$\begin{aligned}H(q)=\cfrac{2(1+q^2)}{1-q+\cfrac{(1+q)(1+q^3)}{1-q^3+\cfrac{2q^2(1+q^4)}{1-q^5+\cfrac{q^3(1+q)(1+q^5)}{1-q^7+\cfrac{q^...
6
votes
0answers
287 views

Does the set of $m \in Max(ord_n(k))$ for every $n$ without primitive roots contain a pair of primes $p_1+p_2=n$?

I have made the following observation: for those n even numbers that do not have primitive roots modulo n ,$Pr(n)$, the set $M(n)$ of those $k$ having a maximum multiplicative order $ord_n(k)$ ...
54
votes
3answers
2k views

Is $29$ the only prime of the form $p^p+2$?

I searched for primes of the form $p^p+2$, where $p$ is prime for a range of $p \le 10^5$ on PARI/GP and found that 29 is the only prime of this form in this range. Questions: $(1)$ Is $29$ the ...
28
votes
2answers
993 views

How to prove $\int_0^\infty J_\nu(x)^3dx\stackrel?=\frac{\Gamma(1/6)\ \Gamma(1/6+\nu/2)}{2^{5/3}\ 3^{1/2}\ \pi^{3/2}\ \Gamma(5/6+\nu/2)}$?

I am interested in finding a general formula for the following integral: $$\int_0^\infty J_\nu(x)^3dx,\tag1$$ where $J_\nu(x)$ is the Bessel function of the first kind: $$J_\nu(x)=\sum _{n=0}^\infty\...
37
votes
3answers
1k views

Prove that there exist infinitely many integers $(n^{2015}+1)\mid n!$

I conjecture that there exist infinitely many integers $n$ such that $$(n^{2015}+1)\mid n!.$$ I have seen a simpler problem that there exist infinitely many integers $n$ such that $(n^2+1)\mid n!$. ...
42
votes
2answers
1k views

a conjectured continued fraction for $\tan\left(\frac{z\pi}{4z+2n}\right)$

Given a complex number $\begin{aligned}\frac{z}{n}=x+iy\end{aligned}$ and a gamma function $\Gamma(z)$ with $x\gt0$, it is conjectured that the following continued fraction for $\displaystyle\tan\left(...
24
votes
1answer
777 views

A conjectured continued fraction for $\phi^\phi$

As I previously explained, I am a "hobbyist" mathematician (see here or here); I enjoy discovering continued fractions by using various algorithms I have been creating for several years. This morning, ...
29
votes
4answers
955 views

Does this conjecture about prime numbers exist? If $n$ is a prime, then there is exist at least one prime between $n^2$ and $n^2+n$.

I made an observation on prime numbers, want to check if any conjecture already exist or not? I am a computer programmer by profession and I am interested in number theory. As like many others I am ...
10
votes
2answers
2k views

how to prove this extended prime number theorem?

A Generalized Prime Number Theorem? Conjecture Let $n$ and $k$ be positive integers with $n - 50 > k^2 > 0$ and $n$ sufficiently large. Then for the odd primes we have, when $p$ is the biggest ...
9
votes
1answer
193 views

How to show an infinite number of algebraic numbers $\alpha$ and $\beta$ for $_2F_1\left(\frac14,\frac14;\frac34;-\alpha\right)=\beta\,$?

(Note: This is the case $a=\frac14$ of ${_2F_1\left(a ,a ;a +\tfrac12;-u\right)}=2^{a}\frac{\Gamma\big(a+\tfrac12\big)}{\sqrt\pi\,\Gamma(a)}\int_0^\infty\frac{dx}{(1+2u+\cosh x)^a}.\,$ There is also $...
7
votes
1answer
420 views

Does the Rogers-Ramanujan continued fraction $R(q)$ satisfy this conjectured infinite series

Given the Rogers-Ramanujan continued fraction $R(q)= \cfrac{1}{1+\cfrac{q}{1+\cfrac{q^2}{1+\cfrac{q^3}{1+\ddots}}}}$ where $q=\exp(2\pi i \tau)$, $|q|\lt1$ for the sake of brevity, let us ...
5
votes
1answer
241 views

conjectured new generating function of fibonacci numbers

I conjecture a new generating function for the fibonacci numbers $F_{n}$. Given,the following conjectured q-continued fraction $$\chi(q)=\cfrac{1}{1+q-\cfrac{(1+q^2)}{1+q^3+\cfrac{q^2(1-q)(1-q^3)}{1+...
4
votes
0answers
140 views

Connecting $\sum\limits_{n=0}^\infty\frac{n!\,(2n)!}{(3n+2)!}$ and $\sum\limits_{n=1}^\infty\frac{1}{n \binom{3n}{n}}$

Define three generalized hypergeometric functions (which differs in the starting numerator in blue): $$A_k =\,_kF_{k-1}\left(\left.\begin{array}{c} 1,\frac{\color{blue}{k-2}}{k-1},\frac{k-1}{k-1},\...
7
votes
1answer
1k views

Analog of Cramer's conjecture for primes in a residue class

Let $q$ and $r$ be fixed coprime positive integers, $$ 1 \le r < q, \qquad \gcd(q,r)=1. $$ Suppose that two prime numbers $p$ and $p'$, with $p<p'$, satisfy $$ p \equiv p' \equiv r \ ({\rm mod}\...
8
votes
2answers
322 views

a conjecture of certain q-continued fractions

Given the squared nome $q=e^{2i\pi\tau}$ with $|q|\lt1$, define, $$\begin{aligned}F(q)=\cfrac{1-q^2}{1-q^3+\cfrac{q^3(1-q)(1-q^5)}{1-q^9+\cfrac{q^6(1-q^4)(1-q^8)}{1-q^{15}+\cfrac{q^9(1-q^7)(1-q^{11})}...
5
votes
1answer
648 views

Euler's totient and divisors count function relationship when $[(\frac{\varphi(n)}{2}+1)\cdot(\frac{\tau(n)}{2}+1)] = n$

I am studying the Euler's totient function $\varphi(n)$ and the divisors count function, $\tau(n)$, also named $d(n)$, and recently opened a question (link here) about the following condition: (1)$\...
5
votes
1answer
795 views

Problems about consecutive semiprimes

I was playing around with semi-prime numbers and I made two conjectures, which are: Given any integer $a$, at least one of $a,(a+1),(a+2)$ or $(a+3)$ is not semi-prime. There are infinitely many ...
3
votes
2answers
498 views

Special case of Pillai's conjecture

Pillai's conjecture is a generalization of Catalan's conjecture. It's say that for fixed positive integers $A, B, C$ the equation $Ax^n - By^m = C$ has only finitely many solutions $(x,y,m,n)$ with $(...
2
votes
1answer
68 views

A number-theory question on the deficiency function $2x - \sigma(x)$

Let $\sigma(x)$ be the sum of the divisors of a (positive) integer $x$. (For example, $\sigma(2) = 1 + 2 = 3$.) Define the deficiency function $D(x)$ to be the number $$D(x) = 2x - \sigma(x).$$ Let ...
5
votes
1answer
393 views

Conjecture about the product of the primitive roots modulo a prime number ($\prod Pr_p$)

While I was learning about the primitive roots modulo $p \in \Bbb P$ (I will call $Pr_p$ to the complete list of the primitive roots module $p$) and having in mind the conjecture explained in this ...
1
vote
0answers
93 views

Is $p(p + 1)$ always a friendly number for $p$ a prime number?

Let $\sigma(x)$ denote the sum of the divisors of $x$. We call the ratio $I(x) = \sigma(x)/x$ as the abundancy index of $x$. A positive integer $N$ is friendly if there exists a positive integer $M \...
1
vote
2answers
65 views

Does “finite average” exist?

I will call an average any continuous function $f(v_1,\dots,v_n)$ of $n$ arguments such that it lies in the closed interval $[\min(v_{1},\dots ,v_{n});\max(v_{1},\dots ,v_{n})]$, is symmetric for all ...
0
votes
2answers
200 views

Conjecture for $\ln(x)$ and $\frac{d^ny}{dx^n}$

I'm not really sure where to start, I found the first, second, third and fourth derivatives of $\ln(x)$ to be $\frac{1}{x}$,-$\frac{1}{x^2}$, $\frac{2}{x^3}$, and -$\frac{6}{x^4}$, respectively. ...
85
votes
2answers
5k 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. Can ...
102
votes
24answers
4k 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 ...
74
votes
18answers
7k views

Conjectures (or intuitions) that turned out wrong in an interesting or useful way

The question What seemingly innocuous results in mathematics require advanced proofs? prompts me to ask about conjectures or, less formally, beliefs or intuitions, that turned out wrong in ...
69
votes
2answers
1k views

Conjecture $_2F_1\left(\frac14,\frac34;\,\frac23;\,\frac13\right)=\frac1{\sqrt{\sqrt{\frac4{\sqrt{2-\sqrt[3]4}}+\sqrt[3]{4}+4}-\sqrt{2-\sqrt[3]4}-2}}$

Using a numerical search on my computer I discovered the following inequality: $$\left|\,{_2F_1}\left(\frac14,\frac34;\,\frac23;\,\frac13\right)-\rho\,\right|<10^{-20000},\tag1$$ where $\rho$ is ...
51
votes
2answers
953 views

How can I prove this closed form for $\sum_{n=1}^\infty\frac{(4n)!}{\Gamma\left(\frac23+n\right)\,\Gamma\left(\frac43+n\right)\,n!^2\,(-256)^n}$

How can I prove the following conjectured identity? $$\mathcal{S}=\sum_{n=1}^\infty\frac{(4\,n)!}{\Gamma\left(\frac23+n\right)\,\Gamma\left(\frac43+n\right)\,n!^2\,(-256)^n}\stackrel?=\frac{\sqrt3}{2\,...
40
votes
2answers
2k views

On Ramanujan's Question 359

In JIMS 4, p.78, Question 359 was asked by Ramanujan. (See The Problems Submitted by Ramanujan to the Journal of the Indian Mathematical Society, p. 9, by Bruce Berndt, et al.) If, $$\sin(x+y) = 2\...
91
votes
11answers
13k views

Is there any conjecture that has been proved to be solvable/provable but whose direct solution/proof is not yet known?

In mathematics, is there any conjecture about the existence of an object that was proven to exist but that has not been explicitly constructed to this day? Here object could be any mathematical object,...
43
votes
2answers
1k views

there exist infinite many $n\in\mathbb{N}$ such that $S_n-[S_n]<\frac{1}{n^2}$

recent conjecture :Let $S_n=1+\frac{1}{2}+\frac{1}{3}+\ldots+\frac{1}{n}$, where $n$ is a positive integer. Prove that :there exist infinite many $n\in\mathbb{N^{+}}$ such that $$S_n-[S_n]<\dfrac{1}...
53
votes
1answer
3k 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 $[...
31
votes
7answers
10k 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 ...
14
votes
3answers
2k 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 ...

1
2 3 4 5