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

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4
votes
1answer
148 views

conjecture about primes and a certain q-series.

Using wolfram Mathematica ,I observed an interesting and surprising property concerning prime numbers and q-series which I could not prove.Yet there is strong evidence supporting it. I would be happy ...
1
vote
1answer
95 views

Can the sum of 3 unique primes be expressed as the sum of 2 primes?

Let's consider the example, $$ 3 + 11 + 19 = 33 \\ 2 + 31 = 33 $$ we can see that there are cases where the sum of 3 primes be expressed as the sum of 2 primes. However, I couldn't find a case ...
1
vote
0answers
72 views

If $(a_n)$ is increasing and $a_n^{1/c^n}\to\infty$, then $\sum\frac1{a_n}$ is irrational?

I $\DeclareMathOperator\lcm{lcm}$am trying to generalise the result from this question: If $(a_n)$ is increasing and $\lim_{n \to \infty} a_n^{1/2^n} = \infty$, show that $\sum_{k=1}^{\infty} 1/{a_n}$ ...
0
votes
1answer
69 views

Improved Betrand's postulate

I want to show that $2p_{n-2} \geq p_{n}-1$... Bertand's postulate shows us that $4p_{n-2}\geq p_{n}$ but can we improve on this? any ideas?
3
votes
1answer
56 views

Sum involving the number of zeros of $k$

I'm currently interested in sums involving digit-functions. Especially, I'd like to calculate the following sum: $$ s=\sum_{k=1}^{\infty}{\frac{a_0(k)}{k\left(k+1\right)}} $$ Where $a_0(k)$ is the ...
12
votes
0answers
163 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\\ ...
1
vote
0answers
35 views

The asymptotic behaviour of triples $n!+q^{n!}=c$, where $q$ is the first prime greater than $n$, and abc conjecture

For a large positive integer $n$, let $q=q(n)$ (below we denote this $q=q(n)$ by $q_{N}$ because we assumed that is the $N$-th prime number) the first prime number which is (strictly) greater that ...
2
votes
0answers
60 views

Beal Conjecture and ($\bmod 3$) operation [closed]

When we apply a ($\bmod 3$) operation on the $A^x +B^y =C^z$ we will see some strange results. For e.g.: Let $A=6m+1$ & $B=6n+1$. Since $A$ & $B$ are odd numbers, $C$ will have to be even. ...
35
votes
3answers
787 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 ...
1
vote
0answers
39 views

Consequences of the Carathéodory conjecture

This is a very stupid question. What are consequences and applications of the Carathéodory conjecture? It seems to me interesting, but completely useless.
1
vote
0answers
70 views

Inequality in 3 variables (conjecture)

Let $a, b, c$ be nonnegative real numbers such that $a+b+c=3$. If $0<k\leq 3+2\sqrt{3}$, then $$\frac{a}{b^2+k}+\frac{b}{c^2+k}+\frac{c}{a^2+k}\geq \frac{3}{1+k}$$ If $k=3+2\sqrt{3}$, then equality ...
2
votes
3answers
378 views

How can I know if my conjecture is not lacking mathematical formality? [closed]

I'm a teenager and student who came up with his own conjecture. Because I'm not a mathematician and I haven't got the knowledge yet, I would like to know if my conjecture doesn't exceed the limits of ...
5
votes
0answers
93 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 ...
6
votes
1answer
64 views

Where $ax + b$ prime infinitely often, is $ax + b - 2$ semiprime at least once?

I'm trying to figure out a way to prove this: Given arithmetic progression $ax + b$ where $a$, $b$ coprime and $ax + b$ is prime infinitely often, it is the case at least once that $ax + b - 2$ is ...
1
vote
2answers
125 views

What do you think about this conjecture?

Beforehand, please know that I'm a little bit of an amateur mathemitician, so this could be very wrong. But, I have tested it over and over and over again and it seems to be plausible, but there's ...
1
vote
1answer
48 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 ...
2
votes
3answers
100 views

Conjecture about linear diophantine equations

I've been dabbling with linear Diophantine equations and came across a rather interesting pattern that I would like to conjecture as true but I have no idea how about to come up with a proof. Let ...
3
votes
0answers
162 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
1answer
78 views

Why are there so many conjectures in number theory and comparatively less in others?

My question is that : Why are there so many conjectures in elementary number theory and comparatively less in others? This is particularly weird because every topic in maths should have its equal ...
1
vote
0answers
63 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
62 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 ...
13
votes
2answers
184 views

Prove that $\lim_{n\rightarrow \infty} \frac{\log_{10}\lfloor\text{Denominator of } H_{10^n}\rfloor+1 }{10^n}=\log_{10} e$

In short, my question is asking to prove that the $$\lim_{n\to\infty}\frac{\text{number of digits in the denominator of} \sum_{k=1}^{10^n} \frac 1k}{10^n}=\log_{10} e$$ I know that the number of ...
6
votes
0answers
216 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 $ord_n(k)$ ...
4
votes
2answers
79 views

Conjecture: only one even Fibonacci term divided by two gives a prime: $F(9) = 34 = 2 \times 17$

Every Fibonacci term $F(3n)$ is divisible by two $F(3) = 2$ $F(6) = 8$ $F(9) = 34$ $...$ After seeking Fibonacci tables factorization until $F(10000)$, for every term $\frac{F(3n)}{2}$, it ...
5
votes
1answer
153 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 ...
-3
votes
1answer
438 views

Is this attempted proof of ABC conjecture correct [closed]

This mathematician claims that he has tackled ABC conjecture! He uses induction and simple inequalities to achieve the result. Is this some serious stuff or is there a basic flaw in the reasoning?
2
votes
3answers
144 views

Why is it that if you square two prime numbers and add them, you get a number that is even and is not a perfect square?

If you do $x^2 + y^2 = n$ where $x$ and $y$ are both prime numbers and are both greater than $3$, why is $n$ always an even number that isn't a perfect square?
11
votes
2answers
305 views

Yet another conjecture about primes

Let $\mathcal{N}(n)$ be the next prime greater than $n$. Conjecture: $\mathcal{N}(n!)-n!\:$ is either $1$ or a prime. It holds for n=1 to 99 and the expression is 1 for 3,11,27,37,41,73,77 and ...
6
votes
2answers
135 views

The conjecture that no triangle has rational sides, medians and altitudes

I have found a conjecture that there is no triangle whose sides, medians, altitudes and area are all rational. I figure that someone must have already found such a triangle if one existed and yet I ...
3
votes
3answers
967 views

Result of solving an unsolved problem?

I was mulling over currently unsolved problems in mathematics (as I, and many others, find them quite interesting) and began to wonder what would happen if these problems were to be solved. I know ...
2
votes
2answers
36 views

Algorithmic procedure to establish the possibility of finding a proof of a conjencture

I was always puzzled by these conjectures which can be stated quite simply, yet finding a proper proof is a monumental task even for the most brilliant mathematicians . Consider the following ...
5
votes
1answer
203 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
vote
2answers
55 views

Prove that $\det(A) \cdot v \, A^{-1} = \det(A+uv) \cdot v \, (A+uv)^{-1}$.

Let $A$ be a $n \times n$ matrix, $u$ a $n \times 1$ matrix and $v$ a $1 \times n$ matrix. If $A$ and $(A+uv)$ are invertible, prove that $$ \det(A) \cdot v \, A^{-1} = \det(A+uv) \cdot v \, ...
1
vote
1answer
103 views

Euler's Totient function $\forall n\ge3$, if $(n-\phi(n)) = \sqrt{n}\ $,$\ $then $(n-\phi(n)) \in \Bbb P$

Recently I opened a question about what it might be a new property of Euler's Totient function. I am still studying the Totient function and I found another interesting relationship, it is very ...
18
votes
2answers
560 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 (link to ...
3
votes
2answers
75 views

Conjectured diagonal Ramsey numbers

While doing some reading on the Wikipedia page for Ramsey numbers, I stumbled upon OEIS sequence A120414, which lists the allegedly conjectured values of the diagonal Ramsey numbers $R(n,n)$. I ...
4
votes
3answers
94 views

Given an odd $x$ there is an $m,n$ such that $2^n + 1 = 3^m x$?

I'm curious about this question: Is it true that for any odd number $x\in 2\mathbb N + 1$ there exists numbers $m,n\in \mathbb N \cup \{0\}$ such that $$2^n+1 = 3^mx$$ Edit: I'm not trying to make ...
0
votes
1answer
51 views

Binary Representation of the Collatz Conjecture

What is the benefit of looking at the binary representation of the collatz conjecture. I know that it makes the computation easier because there is really one operation involved which is multiplying ...
7
votes
2answers
437 views

Trying mathematical induction with $3n+1$ conjecture

Collatz's Conjecture is also known as the $3n+1$ conjecture. Well I thought since the conjecture is dealing with natural numbers so we might as well try mathematical induction and see why it doesn't ...
0
votes
1answer
87 views

A question about the $3n+1$ conjecture

So I know that if you take any even number $n$ that is a power of $2$ like $32 = 2^5,16=2^4$ or $64=2^6$ we will keep dividing by 2 until we reach 1. and so all the steps will be $\frac{n}{2}$ and we ...
2
votes
2answers
90 views

Can I demonstrate a proposition without a rigorous proof?

Writing a paper, I want to declare a certain proposition, but is only supported by other research's empirical result. In this sort of case, what can I name to my argument in my paper? CONJECTURE is so ...
1
vote
1answer
115 views

Asymptotic behavior of the zeros of the digamma function

The gamma function has just one extremum on each interval $(k,k+1)$, where $k$ is a negative integer. These extrema occur at the zeros of the derivative of the gamma function. Let $z_n$ denote the ...
7
votes
1answer
154 views

Power towers of $2$ and $3$ - looking for a proof

Let $\uparrow$ denote the right-associative exponentiation operator: $a\uparrow b\uparrow c=a\uparrow(b\uparrow c)=a^{b^c}$ There is a sequence $A248907$ recently submitted to OEIS (see also ...
0
votes
1answer
27 views

Prove conjecture using premises

I have three premises with me defined: $(B \land L) \implies A$ $(A \land D) \implies \lnot H$ $\lnot J \implies (D \land \lnot H)$ I need to prove the following conjecture with the help ...
0
votes
1answer
118 views

Legendre's Conjecture!

This is my last attempt to the Legendre's Conjecture: based on my first one, it's not that difficult to follow, I'm not using logical manipulation or something like this, it's all about inequalities ...
0
votes
4answers
81 views

In the Collatz function, why does $2^k-1$ reach $3^k-1$ after $2k$ steps, and could it be used to find divergent trajectories?

If you start calculating the Collatz function from an integer of the form $2^k-1$, you will reach $3^k-1$ after $2k$ steps. So, it is possible to pick a starting value that continuously zig-zags ...
1
vote
2answers
48 views

In the Collatz function, why $3^{2k}-1$ and $3^{2k-1}-1$ always share the same trailing trajectory?

Why are the trajectories always the same for numbers of the form $3^{2k}-1$ and $3^{2k-1}-1$ for the Collatz function? For example, let $k = 3$. So, $3^6-1 = 728$ and $3^5-1 = 242$. The trajectories ...
4
votes
3answers
106 views

Generalizing conjecture of Bertrand

Joseph Bertrand's conjecture for primes from 1845, $\;p_{n+1} < 2p_n$, proved by Chebyshev 1852, can be generalized as follows: $$\forall a\in\Bbb Z_+\exists N\in\Bbb Z_+:(n\ge N\implies\exists ...
0
votes
1answer
47 views

Has this partial result about Legendre's Conjecture been proved?

I'd like to know if it has been proved this "partial result" about the Legendre's Conjecture: (1) There are infinitely many $n$ such that there's a prime in $(n^2, (n+1)^2)$ Thanks!
-3
votes
1answer
471 views

$k$-tuple conjecture.

This conjecture is false. See this post Time is running out Suggest notation for Steps 1 and 2. Earn the bonus. For each $k\in\mathbb{Z^{+}}$. Step 1: Create a list $(1,1,1,1...,1)$ of length ...