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

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36
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1k 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 ...
15
votes
0answers
467 views

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

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 ...
12
votes
0answers
167 views

A conjectured identity for tetralogarithms $\operatorname{Li}_4$

I experimentally discovered (using PSLQ) the following conjectured tetralogarithm identity: $$\begin{align}&\phantom{+\;}19\!\;\pi^4-570\ln^42-90\ln^43\\ ...
11
votes
0answers
313 views

Are all totient values of Fibonacci Numbers distinct?

This question was inspired while I was seeing how certain recurrence relations would behave when I applied Multiplative Functions. Let $F_{n}$ be a sequence for which $F_{1}=1,F_{2}=1$, and ...
9
votes
0answers
330 views

Asymptotic FLT $\implies$FLT using ABC Conjecture

Edit: I'm beginning to suspect I either misread the sources or perhaps something wasn't stated, but it's my guess now that there is no nontrivial way of showing the ABC conjecture implies Fermat's ...
8
votes
0answers
133 views

Mathematical conjectures for which Numerical Search promising

I am fascinated by mathematical conjectures, especially those that are believed to be false. Now I wonder, are there some mathematical conjectures for which the numerical search would be promising? ...
6
votes
0answers
120 views

Limit superior, limit inferior and a series involging $\sum_{k\nmid n}$k, where $1\leq k\leq n$

The purpose of this post is state assertions by the use of statements and hypothesis in an expository way and after I am asking for reasonable unconditionally results that you can provide us. Using ...
6
votes
0answers
219 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 $ord_n(k)$ ...
5
votes
0answers
97 views

What steps have been taken so far to solve Brocard's Problem?

The equation is $$n!+1=m^2$$ where $n$ and $m$ are natural numbers. Brocard's Problem asks whether there are solutions for n other than $4, 5, 7$. The only improvement I have found that people have ...
5
votes
0answers
125 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 ...
4
votes
0answers
26 views

Are there infinitely many primes in any sequence determined by a $k$ that is not a Sierpinski number?

Consider the sequence of numbers ranging over $n$ of the form $k\cdot a^n + 1$ for a fixed, odd natural number $k$. The number $k$ is considered a Sierpinski number if the sequence determined by that ...
4
votes
0answers
40 views

If $|G_1|=|G_2|<\infty$ and $|G_1'|<|G_2'|$, then $|Z(G_1)|\geq |Z(G_2)|$? where $G'$ is the commutator subgroup of $G$.

We know that $G'$ characterization how ``abelian'' of a group because we have a theorem: if $G'=\{e\}$, then $G$ is abelian. I have a conjecture. If there are two finite groups $G_1$ and $G_2$, ...
4
votes
0answers
56 views

Why Navier-Stokes solutions are expected to be $C^\infty$?

I am curious, why the Millennium problem about Navier-Stokes is about smooth ($C^\infty$) not just $C^1$ solutions?
4
votes
0answers
113 views

A mixture with ingredients of two equivalences with Riemann Hypothesis

Let $f(x)=x\cdot(\log x)^x$ for $x\geq 2$, then integrating $\log f(x)=\int_2^x f'(t)/f(t)dt$, it is easy to prove the first statement of following, and directly if we put $x=e^H_n$ and add $H_n$ the ...
4
votes
0answers
450 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
118 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
280 views

The Divisors of $s(2s+1)$ and Primes $2n+1$ and $3n+1$ part 1

I want to check my math (and proof) on the following claim. The claim is by way of a computer search and a "hunch". claim: If $s$ is a prime number I write $\varphi_{s} =s(2s+1)$. Let $\tau$ be ...
3
votes
0answers
72 views

How close are we to knowing the rate of convergence to $0$ of $\prod_{p\le x}(1-1/p)^{-1}-e^\gamma\log x $?

This is a question related to an earlier one of mine, which I may answer myself eventually, as I have learnt more about the topic. Despite what one can read on the MathWorld page about Mertens' third ...
3
votes
0answers
64 views

$(\sum_i a_i^2)(\sum_i b_i^2)+(\sum_ia_i b_i)^2\geq \sqrt{(\sum_i a_i^4)(\sum_i b_i^4)}+\sum_ia_i^2b_i^2$?

I have no idea about how to prove (or disprove) the following inequality: $$ \left(\sum_{i=1}^n a_i^2\right)\left(\sum_{i=1}^n b_i^2\right)+\left(\sum_{i=1}^na_i b_i\right)^2\geq ...
3
votes
0answers
46 views

Does the antipode of a f.d. Hopf algebra have finite order

In a lecture I have heard that the antipode of a finite-dimensional Hopf algebra is conjectured to be finite, and that this has only been proven in characteristic $0$ by Larson and Radford in 1988. I ...
3
votes
0answers
109 views

Combining Firoozbakht's conjecture and abc conjecture

Firoozbakht's conjecture states that for all $n\geq 1$ $$p_n^{\frac{1}{n}}>p_{n+1}^{\frac{1}{n+1}},$$ where $p_k$ the kth prime number. By asumption of this conjecture, for a fixed $n$, there is a ...
3
votes
0answers
89 views

Twin primes sums conjecture

I have found an interesting conjecture between twin primes sums. I don't know if it is already described by someone else. I have checked in internet, but I didn't find any mention of such conjecture. ...
3
votes
0answers
81 views

Up to which value is Rassias' conjecture verified?

I came across this conjecture: Rassias' conjecture Up to which $p$ has this conjecture be verified ? Are there intermediate results related to this conjecture ? The conjecture can be ...
3
votes
0answers
96 views

conjectured identity of the product of two theta functions

Looking into the discussion in this post,I was naturally led to consider the following general identity Given the two jacobi theta functions,$$\theta_2(q)=\sum_{n=-\infty}^\infty q^{(n+1/2)^2}$$ and ...
3
votes
0answers
51 views

Can the determinant of an integer matrix with $k$ given rows be the gcd of the determinants of the $k\times k$ minors of those rows?

I'm interested if the following is true: Let $n\geq k\geq1$ be integers, let $A\in\mathbb Z^{k\times n}$ and denote the $\binom nk$ $k\times k$ minors of $A$ by $A_1,\ldots,A_N$. Then the ...
3
votes
0answers
166 views

Very tentative proof of Beal's Conjecture?

I'm a high school student, so please point out my mistakes nicely and in layman's terms :) Thanks! Ok. Beal's Conjecture: If $$a^x+b^y=c^z$$ where $a$, $b$, $c$, $x$, $y$, $z$ are whole numbers; $x, ...
3
votes
0answers
89 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 $\max[x,y] \le n$.
3
votes
0answers
90 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 ...
3
votes
0answers
83 views

irreducibility conjecture

I tried to prove that every polynomial of the form $f(m,n) := m\cdot x^{n-m}+(m+1)\cdot x^{n-m-1}+\cdots+(n-1)\cdot x+n \quad \text{with} \quad 0 < m < n$ is irreducible over the rationals for ...
3
votes
0answers
89 views

Prime clasfication by some constructive function

How to prove or justify the following: $$ f(g)= \frac1{1-g^2} \prod_{k=1}^{\infty} \left(\frac{\sin(\pi \frac gk)}{\pi \frac gk} \cdot \frac 1{1-\frac{g^2}{k^2}}\right), $$ The above statment can ...
2
votes
0answers
125 views

A conjecture about primes

Let $p_n$ be the $nth$ prime and define $p_n^{(m)}$ by $p_n^{(1)}=p_n$ and $p_n^{(m+1)}=p_{p_n^{(m)}}$: $p_n^{(2)}=p_{p_n}$, $\;p_n^{(3)}=p_{p_{p_n}}$ and so far... For some coprime numbers $a,b$, ...
2
votes
0answers
39 views

Number of primes of a certain form

Let $p_n$ be the nth prime. Are there an infinite number of primes of the form $2p_n+1$? Is something known about questions like this?
2
votes
0answers
172 views

A Conjecture Sharper than Cramér's and Firoozbakht's

Notation: $\lfloor\cdot\rfloor$ is floor function; and $\pi(x)$ is the prime-counting function up to $x$. $g_k := p_{k+1} - p_k$ . OEIS sequence A267549 is "Primes prime(k) such that floor( ...
2
votes
0answers
46 views

Does the Descartes number $D = 198585576189$ have a friend?

Let $\sigma(X)$ be the sum of the divisors of $X$, and denote the abundancy index $\sigma(X)/X$ by $I(X)$. If the equation $I(X) = r/s$ has no solution $X \in \mathbb{N}$, then $r/s$ is said to be an ...
2
votes
0answers
37 views

Is always two times an even semiprime at a distance $1$ or prime to the closest previous odd semiprime?

This is an observation regarding the semiprimes, also named 2-almost primes, biprimes, or the product of two primes. This week I do not have a computer, only a tablet (hospitalized with a lot of free ...
2
votes
0answers
77 views

Fibonacci Numbers and the Harmonic Series

$$\sum_{k=1}^{n} \frac{1}{k}=H_n=\frac{p_n}{q_n}$$ Where $p_n,q_n$ are coprime intergers. The first few values for $p_n+q_n$ are $2,5,7,37,197,69,504,1041,9649$. When are $p_n+q_n$ Fibonacci ...
2
votes
0answers
77 views

A false conjecture by de Polignac

(This question would be similar to my other on Goldbach's conjecture so I'll change the "rules") In 1848 de Polignac claimed that "every odd integer is the sum of a prime $p$ and a power of $2$". For ...
2
votes
0answers
79 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 ...
2
votes
0answers
164 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 ...
2
votes
0answers
115 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 ...
2
votes
0answers
57 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
0answers
95 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 ...
2
votes
0answers
54 views

How to prove that every $l$ (such that $ 2 \leq l \leq \lfloor \sqrt{k^2+2n+1} \rfloor $) divides at least one of the following numbers?

$ k^2+2n, k^2, k^2+1, 2n, 2n+1$, (for some $n$) if $k$ is even and $0 < n < k$. I have no idea of how to prove that. I'm working on Legendre's conjecture. Update 1: Yes, for $n=0$ all $l$ ...
2
votes
0answers
218 views

Minimum amount of primes between squared primes

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
0answers
170 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 ...
2
votes
0answers
438 views

Can the openness of Gilbreath's conjecture be reduced to proof of the Riemann hypothesis?

As an information theoretician, it's a personal hobby of mine to find elegant analogies to open mathematical problems. After all, they have a profound impact on how I conduct my research and how I ...
1
vote
0answers
21 views

Factorial and primorial twin primes

Factorial primes are are primes of the form $n! \pm 1$ and primorial primes are primes of the form $p\#\pm 1$, where $p\#$ is the product of all primes $\leq p$. To cite ...
1
vote
0answers
53 views

$\pi\left(\left(n+m\right)^2\right) - \pi\left(n^2\right) \ge 2 \cdot m$

Conjecture For $n \ge 1 $ , $m \ge 1$ $\pi\left(\left(n+m\right)^2\right) - \pi\left(n^2\right) \ge 2 \cdot m$ where $\pi\left(n\right)$ is the prime counting function . Does this conjecture ...
1
vote
0answers
30 views

The Divisors of $s(2s+1)$ and Primes $n$, $4n+1$, and $6n+1$

This question is somewhat related to this one. Most of this is by way of a computer search: claim: If $s$ is any positive integer I write $\varphi_{s} =s(2s+1)$. Let $\tau$ be the divisor ...
1
vote
0answers
24 views

Can we capture the definition to be a Mersenne prime in an identity only involving the arithmetic function $S(n)=\sum_{k=1}^n\text{nmod k}$?

Let $S(n)=\sum_{k=1}^n\text{nmod k}$ the sum of remainder function, denoting $\sigma(n)$ as the sum of divisors function, it is know that for each $n>1$ $$S(n)-S(n-1)=2n-1-\sigma(n),$$ is as ...