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

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3
votes
2answers
73 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 ...
2
votes
2answers
205 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 ...
8
votes
1answer
108 views

Question regarding the status of Erdős' conjectures

Has Erdős' conjecture on arithmetic progressions or his conjecture on Sylvester's sequence been proven?
2
votes
1answer
72 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 ...
1
vote
1answer
93 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 ...
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 ...
21
votes
0answers
580 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 ...
8
votes
0answers
295 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 ...
5
votes
0answers
75 views

Possibly New Prime Conjecture

I was in the midst of proving a conjecture when I came across an observation that led me to forming a potentially new conjecture. The conjecture goes as follows: Any given sum of twin primes ...
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 ...
4
votes
0answers
42 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 ...
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
93 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
109 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 ...
3
votes
0answers
83 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
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 ...
3
votes
0answers
81 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
84 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 ...
3
votes
0answers
365 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 ...
2
votes
0answers
76 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
105 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
43 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
94 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
51 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
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 ...
1
vote
0answers
24 views

Is the conjecture $Per A $ is the largest eigenvalue of $\tilde{A}$ being solved?

Let A be a positive semidefinite matrix of order $n$. Is the conjecture $Per A $ is the largest eigenvalue of $\tilde{A}$ is being solved? Where $\tilde{A}$ is the matrix of order $n!\times ...
1
vote
0answers
54 views

How I can make a proof to this conjecture if it is possible?

Is there someone who can show me the way helping me to proof this conjecture. If it's not open, at a least show me links or papers which to be helpful. Conjecture: Assume $c > 0$ and that an ...
1
vote
0answers
25 views

A question and a conjecture on $USp(N)$ group

$USp(N)$ with $N$ an even integer is defined as the group of unitary matrices $M$ that satisfy $M^TJM=J$, where $M^T$ is the transpose of $M$ and $J$ is the anti-symmetric $N$-by-$N$ matrix ...
1
vote
0answers
108 views

Conjecture: for even n without primitive roots modulo n, the set of $m \in Max(ord_n(k))$ contains one pair of primes $p_1+p_2=n$ (Goldbach)

Conjecture: 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 $ord_n(k)$ contains at least a pair of primes ...
1
vote
0answers
102 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 ...
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}$, $\:$ ...
0
votes
0answers
29 views

Addressing a conjecture with different “strengths”

I plan on writing a conjecture in a publication I am developing. However, the conjecture has different "strengths" to it, take this example: Conjecture 1.1. $\exists a$, such that $a$ is a solution ...
0
votes
0answers
49 views

How to convert an object into a sphere?

I'm not sure if I understand it enough, but doesn't the Poincare conjecture prove any shape can be turned into a sphere? How would I go about transforming such an object? Like let's say I have a ...
0
votes
0answers
20 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 ...
0
votes
0answers
125 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 $\ ...
0
votes
0answers
32 views

A conjecture on the product of digits of a number

Define $(m,n)$ to be a special pair if $n=m \cdot Pd(n)$. Where $Pd(n)$ is the product of digits $n$. Then I have the following conjecture - For every $m$ with no digit of $m$ being $0$ , there ...
-1
votes
0answers
47 views

Very tentative proof that the terms in Beal's Conjecture must not be squares?

I'm a high-school student, so please point out my mistakes accordingly. Thanks! Alright. So: $$a^x + b^y=c^z$$ And if x, y, and z are over 2 then $$a^2a^m+b^2b^n=c^2c^o$$ m, n, and o of course ...