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5
votes
2answers
104 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 ...
2
votes
1answer
43 views

Twin Prime conjecture current status

Can someone help me with a link to read about the status of the Twin Prime conjecture. I have browse on the internet and have read some articles but still I have no clue of the updated status of Twin ...
3
votes
1answer
48 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 ...
-12
votes
1answer
198 views

Is this proof for the Goldbach Conjecture? [closed]

Let's prove the conjecture by induction. Claim: For every even number $n≥4$, there exist primes p and q such that p+q=n. Base case: $n=4$. Let $p=q=2$. Induction step: Say that we know that the ...
1
vote
3answers
35 views

I have a conjecture on local max/min , can any of you propose a contradiction?

If $f$ is a non-piecewise function defined continuous on an interval $I$, and within that interval $I$, there exists a value $x$, such that $f`(x)$ (derivative of $f$) does not exist , then at that ...
4
votes
1answer
280 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 ...
8
votes
4answers
977 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 ...
6
votes
2answers
185 views

About the hyperplane conjecture.

I have recently heard about the hyperplane conjecture and I would like to understand better the problematic behind this conjecture. The hyperplane conjecture: There exists a universal constant ...
21
votes
2answers
442 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 ...
2
votes
1answer
55 views

Srinivasa Ramanujan conjectures

I searched internet for the whole list of conjectures by Srinivasa Ramanujan , but its not fruitful . I came to know that recently a book of Ramanujan was out and contains many conjectures related to ...
3
votes
2answers
47 views

Prove that $\lim\limits_{x\to\infty} \frac{\Gamma(x+1,x(1+\epsilon))}{x\Gamma(x)}=0$.

I conjecture that for any $\epsilon>0$, we have $$ \lim_{x\to\infty} \frac{\Gamma(x+1,x(1+\epsilon))}{x\Gamma(x)}=0 $$ where $\Gamma(x,a) = \int_a^\infty t^{x-1}e^{-t} \mathrm{d}t$ denotes the ...
3
votes
3answers
136 views

Conjectured closed form of $G^{2~2}_{3~3}\left(1\middle|\begin{array}c1,1;b+1\\b,b;0\end{array}\right)$

In my answer to this question, I come across the following case of the Meijer G-function: $$F(b)=G^{2~2}_{3~3}\left(1\middle|\begin{array}c1,1;b+1\\b,b;0\end{array}\right), b>0$$ and based on my ...
2
votes
2answers
547 views

Disprove the Twin Prime Conjecture for Exotic Primes

The List of unsolved problems in mathematics contains varies conjectures of exotic primes like: Mersenne primes (of the form $2^p - 1$ where $p$ is a prime, A000668, $43\%$) Sophie Germain primes ...
16
votes
4answers
433 views

Closed form for $\int_{-1}^1\frac{\ln\left(2+x\,\sqrt3\right)}{\sqrt{1-x^2}\,\left(2+x\,\sqrt3\right)^n}dx$

I'm trying to find a closed form for the following integral: $$\mathcal{J}(n)=\int_{-1}^1\frac{\ln\left(2+x\,\sqrt3\right)}{\sqrt{1-x^2}\,\left(2+x\,\sqrt3\right)^n}dx\tag1$$ I have conjectured values ...
16
votes
2answers
443 views

A conjectured closed form of $\int_0^\infty\frac{x-1}{\sqrt{2^x-1}\ \ln\left(2^x-1\right)}dx$

Consider the following integral: $$\mathcal{I}=\int_0^\infty\frac{x-1}{\sqrt{2^x-1}\ \ln\left(2^x-1\right)}dx.$$ I tried to evaluate $\mathcal{I}$ in a closed form (both manually and using ...
4
votes
1answer
84 views

Conjecture about integral $\int_0^1 K\left(\sqrt{\vphantom1x}\right)\,K\left(\sqrt{1-x}\right)\,x^ndx$

I'm interested in the following integral: $$\mathcal J(n)=\int_0^1 K\left(\sqrt{\vphantom1x}\right)\,K\left(\sqrt{1-x}\right)\,x^ndx,\tag1$$ where $K(z)$ is the complete elliptic integral of the 1ˢᵗ ...
1
vote
2answers
42 views

Conjecture: When does $n=ab$, with $a\leq b\leq 2a$?

I conjecture that if this occurs, $a$ and $b$ are unique. Obviously if $n$ is an odd prime, this does not occur, and if $n=a^2$, it does. In any case, what is the set of numbers such that this sort of ...
42
votes
2answers
1k 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 ...
16
votes
0answers
237 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 $$\mathcal{E}=\frac{_3F_2\left(\frac{1}{2},\frac{3}{4},\frac{5}{4};\ 1,\frac{3}{2};\ \frac{3}{4}\right)}{\Pi\left(\frac{1}{4}\Big|\frac{1}{\sqrt{3}}\right)},$$ ...
32
votes
2answers
4k views

Symmetry of bicycle-lock numbers

Suppose you have a combination bicycle lock of this sort: with $n$ dials and $k$ numbers on each dial. Let $m(n,k)$ denote the minimum number of turns that always suffice to open the lock from any ...
19
votes
2answers
372 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
vote
1answer
34 views

Mertens conjecture - bounds

The disproven Mertens Conjecture states that $$|M(n)|\leq \sqrt{n}$$ If it is bounded at all, would the bounds $$|M(n)|\leq \sqrt{2n\log(\log (n))}$$ not be more realistic, and still consistent with ...
5
votes
0answers
67 views

Mertens conjecture & Riemann hypothesis [closed]

The Mathworld page on the Mertens Conjecture states that $$\limsup_{n\rightarrow\infty}|M(n)|n^{-1/2}=\infty$$ seems very probable (Odlyzko and te Riele 1985). Would it not then follow that ...
0
votes
1answer
135 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 ...
1
vote
0answers
68 views

All about a failed conjecture.

Some months ago I made the following conjecture - 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 ...
39
votes
2answers
394 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 ...
5
votes
0answers
96 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
47 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 ...
0
votes
1answer
37 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 ...
4
votes
2answers
122 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 ...
47
votes
2answers
1k views

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

Consider the following integral: $$\mathcal{I}=\int_1^\infty\frac{\operatorname{arccot}\left(1+\frac{2\,\pi}{\operatorname{arcoth}x\,-\,\operatorname{arccsc}x}\right)}{\sqrt{x^2-1}}dx,$$ where ...
4
votes
2answers
226 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 ...
0
votes
0answers
15 views

Make a conjecture that $g(x)=\lfloor{kx}\rfloor$ for a particular value of k

Make a conjecture that $g(x)=\lfloor{kx}\rfloor$ for a particular value of k. I am thinking that it has to do with where k is a fraction or not but I am unsure how to write it out.
7
votes
1answer
158 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
2answers
78 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
72 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 ...
1
vote
1answer
55 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
80 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 ...
4
votes
0answers
122 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 ...
16
votes
2answers
377 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 ...
3
votes
0answers
330 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 ...
25
votes
1answer
299 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 ...
0
votes
1answer
38 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
0answers
95 views

Is this a satisfiable proof (hypothetical) -yet simple one- of Goldbach's conjecture? [closed]

Last night I found a paper by a guy named Miles MATHis that claims to simply prove Goldbach's conjecture. Abstract: Here I solve Goldbach's Conjecture by the simplest method possible. I do this by ...
2
votes
4answers
115 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\} $
2
votes
0answers
150 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?
35
votes
1answer
948 views

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

$$\int_0^1\frac{dx}{\sqrt{1-x}\ \sqrt[4]x\ \sqrt[4]{2-x\,\sqrt3}}\stackrel?=\frac{2\,\sqrt2}{3\,\sqrt[8]3}\pi\tag1$$ The equality numerically holds up to at least $10^4$ decimal digits. Can we ...
1
vote
0answers
105 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
2answers
113 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?
4
votes
1answer
146 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} - ...