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

learn more… | top users | synonyms

0
votes
1answer
153 views

For all $n$, $9^n + 25^n - 1$ has a prime factor with $7$ in its decimal representation?

Let $x_n$ be a sequence of positive integers defined by $x_n=9^n + 25^n -1$ for all $n \ge 2$ I conjectured that there exists at least one prime divisor of $x_n$ which contains $7 $ in its decimal ...
5
votes
0answers
115 views

Are $ut + 1$ and $ut + t + 1$ both prime for some t for any $u$?

Conjecture : For any natural number $u$, there is a natural number $t$ such that $ut + 1$ and $ut + t + 1$ are both prime. So we get a solution of the equation $$au - b(u+1) = -1$$ with prime ...
0
votes
1answer
46 views

Proof envolving center of a group and conjugacy classes

I need to prove that every group $G$ of order 9 is abelian. So, this is what I've done so far: There are three options for $[G:Z(G)]$: $[G:Z(G)]=1$, in this case we are done, since $G=Z(G)$ and that ...
0
votes
1answer
58 views

Conjecture linking multiplicative order of $2$ and semi-primes

Suppose we have a semi-prime $N=pq$, where $p \ne q$, and $p>2$, $q>2$ Let $k$ be the multiplicative order of $2$ mod $N$, then either $p^{2} \bmod k \equiv 1$ or $q^{2} \bmod k \equiv 1$ Is ...
4
votes
2answers
147 views

Conjecture involving semi-prime numbers of the form $2^{x}-1$

Let $x$ be a positive integer such that $(2^{x}-1)=pq$ , where $p$ and $q$ are prime numbers. I want to show that either $p^{2} \bmod x \equiv 1$ or $q^{2} \bmod x \equiv 1$ (or both of course). Is ...
4
votes
2answers
255 views

A conjecture on $\phi(n)$

Let $\phi(n)$ denote the Euler totient function of $n $. Then let $N$ be a number such that $\phi(N)$ divides $N$ . Also let $I_1= \frac{N}{\phi(N)}$ which is defined as the "Second order Index of ...
8
votes
1answer
207 views

Proving infinitude of primes in a certain form.

Here I have the following conjecture -Let $$S_1(n)= \frac{(n-1)! +1}{n}$$ then there exist infinite prime numbers $p$ for which $S_1(p)$ is prime. And I don't know how to prove it. EDIT Let ...
2
votes
1answer
131 views

Proof on a conjecture involving $d(N)$

Let $d(n)$ denote the number of divisors of $n $. Then let $N$ be a number such that $d(N)$ divides $N$ . Also let $I= \frac{N}{d(N)}$ which is defined as the "Index of Beauty of $N$ ". Then prove ...
2
votes
2answers
140 views

An approach to Andrica's conjecture

Andrica's conjecture states that $\sqrt{p_{n+1}}-\sqrt{p_n} < 1$. but solving for $n=1,2,\dotsc$ yields n=1, $\sqrt{p_{2}}-\sqrt{p_1} < 1$=>$\sqrt{p_{2}}<\sqrt{p_1}+1$ n=2, ...
1
vote
1answer
78 views

If AC is false , is this statement about the halting problem true?

Assume AC is false. (AC = axiom of choice ) Let $n,m$ be positive integers. Let $f: \Bbb N \rightarrow \Bbb N$ and $f(m)=m$. Let $g(n,m)=1$ if the iterations $f(n),f(f(n)),...$ converges to $m$. ...
1
vote
2answers
81 views

Why is $2^a > a^3$?

I found this rather interesting and maybe, a bit too obvious for some people property about 2 raised to some power. $2^a > a^3$, if $a=0,a=1 \text{ or } a\ge 10$ .($a \in N$) I seem to get a bit of ...
6
votes
1answer
187 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 ...
49
votes
2answers
558 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 ...
25
votes
1answer
346 views

Conjectured closed form for $\int_0^1x^{2\,q-1}\,K(x)^2dx$ where $K(x)$ is the complete elliptic integral of the 1ˢᵗ kind

I am interested in a general closed-form formula for integrals of the following form: $$\mathcal{J}_q=\int_0^1x^{2\,q-1}\,K(x)^2dx,\tag0$$ where $K(x)$ is the complete elliptic integral of the 1ˢᵗ ...
21
votes
3answers
450 views

An integral $\int_0^\infty P_s(x-1)\,e^{-x}\,dx$ involving Legendre functions

Let $P_s(x)$ denote the Legendre functions of the $1^{st}$ kind, i.e. the Legendre polynomial generalized to an arbitrary (not necessarily integer) order $s$. It can be expressed using the ...
-1
votes
1answer
74 views

Lines formed from vertices of n-gons equate to triangular numbers.

Noticed something neat tonight! The number of unique lines you can form by connecting the vertices of an n-gon is equal to the (n-1)th triangular number. (e.g. in a square all 4 veritices make 4 ...
2
votes
4answers
139 views

is this conjecture true or false?

I want to know if this conjecture istrue or false $$\Large e^{\frac{ \ln x}{x}} \notin \mathbb{Z} $$ for every $x \in \mathbb{R} \setminus \{1,-1,0\} $
8
votes
2answers
379 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?
21
votes
2answers
519 views

Conjecture: $\int_0^1\frac{3x^3-2x}{(1+x)\sqrt{1-x}}K\big(\!\frac{2x}{1+x}\!\big)\,dx\stackrel ?=\frac\pi{5\sqrt2}$

$$\int_0^1\frac{3x^3-2x}{(1+x)\sqrt{1-x}}K\left(\frac{2x}{1+x}\right)\,dx\stackrel ?=\frac\pi{5\sqrt2}$$ The integral above comes from the evaluation of the integral ...
1
vote
0answers
186 views

Proving there exists prime numbers between the squares of prime numbers

Conjecture: $\forall $ $p_{n}$, $p_{n+1} \in \mathbb{P}$, $\:$ $p^2_{n+1} = p^2_{n} +\omega_{n} p_{n} + \phi_{n} : \phi_{n} , \omega_{n} \in \mathbb{N} $ and $ \phi_{n} < p_{n}$, $\:$ ...
2
votes
1answer
128 views

Does there exist a prime number within the interval?

Conjecture $\forall p_{n}\in \mathbb{P} : n\geq3, \: \exists p_{m}\in \mathbb{P} : 3p_{n} - 4 \geq p_{m} > \sqrt{2(p^2_{n+1} - 1)} $ How would you go about proving/disproving this?
5
votes
1answer
184 views

A conjecture about the difference between consecutive primes with respect to a prime number squared.

Conjecture If we have two consecutive prime numbers $p_{a}$ and $p_{a+1}$, and two other consecutive primes $p_n$ and $p_{n+1}$, so that $p_{a} < p_{a+1} < p^2_{n+1}$, then $p_{a+1} - ...
1
vote
1answer
136 views

The name and proof of a conjecture on prime intervals

Conjecture: There exists at least one prime number $p_{m}$ : $ap_{n} < p_{m} < (a+1)p_{n}$, $\forall$ $a \in \mathbb{N}$ and $\forall$ $p_{n}$ $\in \mathbb{P} $ if $(a+1)p_{n} < ...
49
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. ...
53
votes
2answers
2k 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 $$\alpha=\sqrt{6}\ \sqrt{12+7\,\sqrt3}-3\,\sqrt3-6.\tag1$$ Note that $\alpha$ is the unique positive root of the polynomial equation $$\alpha^4+24\,\alpha^3+18\,\alpha^2-27=0.\tag2$$ Now consider ...
3
votes
1answer
286 views

My conjecture on almost integers.

Here when I was studying almost integers , I made the following conjecture - Let $x$ be a natural number then For sufficiently large $n$ (Natural number) Let $$\Omega=(\sqrt x+\lfloor \sqrt x ...
8
votes
1answer
159 views
4
votes
1answer
87 views

What's wrong with my conjecture?

I was doing math homework, and I formulated the following conjecture from one of the questions: If $f(x)$, $g(x)$ and $h(x)$ are continuous functions and the equations $f(x) = h(x)$ and $g(x) = h(x)$ ...
4
votes
0answers
416 views

Totient summatory function

Let $\Phi(n) = \sum_{k=1}^n \phi(k)$ be the totient summatory function. Here is an interesting conjecture I've made: The ratio $\Phi(n^2)/\Phi(n)$ is an integer only for $n=1,2,3,5$ and $6$. I made a ...
4
votes
0answers
113 views

What are the recent advancements in mathematics that an undergraduate can understand?

I am an undergraduate student of mathematics. I am interested to know the conjectures that are proved in the last, say 5 or 10 years. Or any other development. Preferably in pure mathematics. One of ...
3
votes
0answers
77 views

A challenging non homogenous fractional inequality.

The following problem is a challenging generalization of several difficult inequalities, where none of the usual methods used in inequalities seems to work. I would like to know if someone has a ...
12
votes
2answers
342 views

Conjecture $\int_0^1\frac{\ln\left(\ln^2x+\arccos^2x\right)}{\sqrt{1-x^2}}dx\stackrel?=\pi\,\ln\ln2$

$$\int_0^1\frac{\ln\left(\ln^2x+\arccos^2x\right)}{\sqrt{1-x^2}}dx\stackrel?=\pi\,\ln\ln2$$ Is it possible to prove this?
23
votes
1answer
453 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 $$\mathcal{E}=\frac{_3F_2\left(\frac{1}{2},\frac{3}{4},\frac{5}{4};\ 1,\frac{3}{2};\ \frac{3}{4}\right)}{\Pi\left(\frac{1}{4}\Big|\frac{1}{\sqrt{3}}\right)},$$ ...
57
votes
2answers
2k 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 ...
8
votes
1answer
238 views

Integral $\int_0^\infty\frac{dx}{\frac{x^4-1}{x\cos(\pi\ln x)+1}+2x^2+2}$

I need your help with this integral: $$\int_0^\infty\frac{dx}{\frac{x^4-1}{x\cos(\pi\ln x)+1}+2\,x^2+2}.$$ I wasn't able to evaluate it in a closed form, although an approximate numerical evaluation ...
7
votes
1answer
228 views

A conjecture $\int_{-\infty}^\infty\frac{\arctan e^x}{\cosh x}\cdot\frac{\tanh\frac{x}2}{x}dx\stackrel?=\frac\pi2\ln2$

I need to find a closed form for this integral: $$\mathcal{I}=\int_{-\infty}^\infty\frac{\arctan e^x}{\cosh x}\cdot\frac{\tanh\frac{x}2}{x}dx.$$ A numerical integration results in an approximation ...
33
votes
2answers
976 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 ...
0
votes
2answers
1k 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 ...
3
votes
1answer
349 views

Weaker Version of “Goldbach's Other Conjecture”

Taken from problem 46 on 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. $9 = 7 + 2 \times 1^2$ ...
2
votes
0answers
162 views

Approximation to $\pi(x)$ conjecture.

A friend conjectured that $\left[\prod_{k=1}^{a_j <\sqrt{x}} \left(1-\frac{1}{a_k}\right)\right] x$ is usually closer to $\pi(x)$ than $\operatorname{Li}(x)$ is for some (fixed) sequence of ...
5
votes
3answers
860 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 ...
2
votes
4answers
52 views

Conjuncting two independent statements

Suppose there are two statements, $A$ and $B$ that are independent. As far as I know one needn't to prove $A$ or $B$ either, it is enough to generate $C = A \land B$, and then proving $C$ shows $A$ ...
1
vote
1answer
105 views

Solving a conjecture by bruteforcing

Say we wanted to check the Beal conjecture ["If $A^x+B^y=C^z$, where $[A, B, C, x, y, z \in N] \wedge [x, y, z \gt 2] \to $ A, B and C must have a common prime factor", from the official site]. What ...
1
vote
3answers
409 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 ...
18
votes
1answer
291 views

Is $K\left(\frac{\sqrt{2-\sqrt3}}2\right)\stackrel?=\frac{\Gamma\left(\frac16\right)\Gamma\left(\frac13\right)}{4\ \sqrt[4]3\ \sqrt\pi}$

Working on this conjecture, I found its corollary, which is also supported by numeric caclulations up to at least $10^5$ decimal digits: ...
20
votes
2answers
524 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 ...
18
votes
2answers
329 views

How to prove $4\times{_2F_1}(-1/4,3/4;7/4;(2-\sqrt3)/4)-{_2F_1}(3/4,3/4;7/4;(2-\sqrt3)/4)\stackrel?=\frac{3\sqrt[4]{2+\sqrt3}}{\sqrt2}$

I have the following conjecture, which is supported by numerical calculations up to at least $10^5$ decimal digits: ...
3
votes
1answer
76 views

$\int_0^1 \Bigl(\mathrm{Li}_2\bigl(\frac{x-1}{x}\bigr)\Bigr)^2 \mathrm{d}x$

This question has some relationship to this integral: Let $\mathrm{Li}_2$ be the dilogarithm. Then, numerically, $$ \int_0^1 \Bigl(\mathrm{Li}_2\bigl(\frac{x-1}{x}\bigr)\Bigr)^2 \mathrm{d}x = ...
4
votes
1answer
375 views

Conjecture regarding trapping rational numbers in some special intervals

Conjecture: Let $b\in\mathbb{N}_{\geq3}$ and $\{x_i\}$ be a collection of $b−2$ rational numbers greater than $1$. Does there always exist a natural number $a$ such that for all $i$ there exists some ...