The conjectures tag has no wiki summary.
4
votes
0answers
38 views
Closed form for $\int_1^\infty\int_0^1\frac{\mathrm dy\,\mathrm dx}{\sqrt{x^2-1}\sqrt{1-y^2}\sqrt{1-y^2+4\,x^2y^2}}$
Consider the following integral:
$$\mathcal{I}=\int_1^\infty\int_0^1\frac{\mathrm dy\,\mathrm dx}{\sqrt{x^2-1}\sqrt{1-y^2}\sqrt{1-y^2+4\,x^2y^2}}.$$
It can be represented as
...
3
votes
1answer
44 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,
$$
\int_0^1 \Bigl(\mathrm{Li}_2\bigl(\frac{x-1}{x}\bigr)\Bigr)^2 \mathrm{d}x
=
...
3
votes
1answer
307 views
Conjecture regarding trapping rational numbers in some special intervals
Conjecture:
Let $b\in\mathbb{N}_{\geq3}$ and $\{x_i\}$ be a collection of $b−2$ rational numbers greater than $1$. Does there always exist a natural number $a$ such that for all $i$ there exists some ...
5
votes
1answer
186 views
Is this number theory conjecture known to be true?
I've been working on proving that there is always a prime between $n$ and $2n$, and also that there is always a prime between $n^2$ and $(n+1)^2$ (Legendre's conjecture).
I believe I've proven those ...
2
votes
0answers
52 views
irreducibility conjecture
I tried to prove that every polynomial of the form
$f(m,n) := m\cdot x^{n-m}+(m+1)\cdot x^{n-m-1}+\cdots+(n-1)\cdot x+n \quad \text{with} \quad 0 < m < n$
is irreducible over the rationals for ...
6
votes
0answers
67 views
Examples of falsified (or currently open) longstanding conjectures leading to large bodies of incorrect results.
In general, the way that modern mathematical research is conducted isn't the way that many would assume is the ideal method of research. That is, mathematics is not the linear progression of ...
0
votes
4answers
88 views
Conjecture to start a proof
In an inductive proof example, my book starts with the identity stating $$\sum\limits_{i=1}^n \sum\limits_{j=1}^i j = \frac{n(n+1)(n+2)}{6}\;.$$ As a sidenote, it says they reached this identity by ...
2
votes
2answers
128 views
Finding a counterexample to a Prime Factorization Conjecture
Let $\mathbb{Z}_{\geq 2}$ be the set of natural numbers starting at 2:
$$\mathbb{Z}_{\geq 2}= \{2, 3, 4, 5,\ldots\}.$$
An natural number's prime factorization is odd if the total number of primes in ...
2
votes
2answers
118 views
Number of subsets of $U$ whose arithmetic mean is integral
Question is :-
$n$ is a positive integer. Call a non-empty subset $S$ of $\{1,2,\dots,n\}$ "good" if the arithmetic mean of
elements of $S$ is also an integer. Further Let $t_n$ denote the ...
1
vote
1answer
203 views
Converse of Collatz Conjecture
How to write a pseudocode program that halts only if the Collatz Conjecture is
false.
Thanks much in advance!!!
6
votes
0answers
69 views
Question regarding the status of Erdős' conjectures
Has Erdős' conjecture on arithmetic progressions or his conjecture on Sylvester's sequence been proven?
2
votes
2answers
312 views
Which notable mathematicans have tried solving the Riemann hypothesis?
I have read that the Riemann hypothesis is the most important open question in mathematics and has been open since 1859. I am wondering which famous mathematicians have actually tried to solve it and ...
1
vote
1answer
159 views
On expressing a square as a sum of two cubes
Given $a,b,c \in \mathbb{N}$ which satisfy the following conditions:
$a^3 + b^3 = c^2$
$ a \neq b$
=-=-=-=-=-=-=-=-=-=-=-=-=-=
EDIT, Will Jagy: The conjecture is that, for a given $c,$ there are ...
2
votes
2answers
308 views
Why is Goldbach's conjecture not included in the millenium prize problems
As we all know, the Goldbach's Conjecture is one one of the oldest and best-known unsolved problems in mathematics.
I was going through some of the attempts made to solve it and got fascinated as to ...
3
votes
0answers
74 views
Prime clasfication by some constructive function
How to prove or justify the following:
$$ 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),
$$
The above statment can ...
0
votes
1answer
107 views
Goldbach conjecture and primes
I need some clarification on (1) Is there any proof to say Mersenne primes $M_p$ are finite or infinite? if there, could you share here.. (2) If Goldbach is conjecture is true, how you can justify the ...
2
votes
2answers
147 views
Fibonacci conjecture: $(F_{n+5})^2 - (F_n)^2 = 3((F_{n+3})^2 - (F_{n+2})^2) + 8 F_{n+2} F_{n+3} $.
So this is the question I have
The Fibonacci sequence is a recurrence system given by $$F_1 = 1, \ F_2 = 1, \ F_{n+2} = F_{n+1} + F_n \qquad (n = 1, 2, 3, \ldots).$$ This question concerns the ...
0
votes
3answers
64 views
Primes + Inetvel + conjecture on primes
a) Can we establish a proof, there exists infinitely many primes of the form $n^2$ + 1. Why is the unit digit of such a prime p always 1 or 7? Is there any reasonable procedure or concept for the fact ...
0
votes
0answers
44 views
Which graphs satisfy this property?
I am currently looking into a conjecture in graph theory, known as the Jackson conjecture (1992). It says the following, in an equivalent formulation to its equivalent statement:
*Conjecture*Every ...
6
votes
0answers
156 views
Asymptotic FLT => 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 ...
4
votes
1answer
130 views
How to prove the equivalence between the two statements of ABC conjecture?
The ABC conjecture stated by wikipedia says the following statements are equivalent:
I.
For $\epsilon>0$, there are finite coprime triple $(a,b,c)$ satisfying $a+b=c$ such that ...
1
vote
1answer
55 views
Integer solutions to $2^x+3^y=9^z+8^w+8^t$
currently in number theory is there a method to solve for integer solutions to equations like $2^x+3^y=9^z+8^w+8^t$?
For example, $2^{13}+3^6=9^3+8^4+8^4$. (obtained using computer brute force)
In ...
1
vote
1answer
89 views
Integer Partition by Counting Repetition : Conjecture ??
I would like to find informations regarding this way of doing Integer Partitions or
this conjecture,
Suppose you have all the ordered partitions of 5:
5
4 1
3 2
2 2 1
3 1 1
2 1 1 1
1 1 1 1 1
Then ...
2
votes
3answers
252 views
Is this statement stronger than the Collatz conjecture?
$n$,$k$, $m$, $u$ $\in$ $\Bbb N$;
Let's see the following sequence:
$x_0=n$; $x_m=3x_{m-1}+1$.
I am afraid I am a complete noob, but I cannot (dis)prove that the following implies the ...
5
votes
2answers
185 views
Is this conjecture on Fibonacci sums correct?
I have the following conjecture, need to know a proof just in case mine is wrong, or the conjecture itself is wrong.
The sum of $k$ distinct Fibonacci numbers can be written in at most $k$ ways as ...
14
votes
1answer
208 views
Density of odd numbers in a sequence relating base 2 and base 3 expansion
Define the function
$$f(4n)=6n+1\\
f(4n+1)=6n+2\\
f(4n+2)=6n+3\\
f(4n+3)=6n+5$$
and the sequence $u_0=2$, $u_{k+1}=f(u_k)$.
Let $d_1\le d_2$ be the lower and upper asymptotic density of odd numbers ...
1
vote
1answer
499 views
Why proof by induction fails for Goldbach's conjecture?
Can anyone clarify why induction method fails for this conjecture?
11
votes
1answer
230 views
The Goldbach Conjecture and Hardy-Littlewood Asymptotic
A source I am reading refers to the Goldbach conjecture (that every even number is the sum of two primes), and then immediately follows with the "Hardy-Littlewood conjecture" that
$\sum ...
2
votes
2answers
201 views
Question about the Collatz Conjecture, Nicomachus's Triangle, and more
Using $(3p+1)/2$ starting with $p = 44102911$, we find an ordered set of $8$ primes. By computer, we find that this is the only ordered set of $8$ primes $< 300000000$ primes.
The primes:
...
0
votes
1answer
291 views
Disprove the Twin Prime Conjecture for Exotic Primes
The List of unsolved problems in
mathematics
contains varies conjectures of exotic primes like:
Mersenne primes (of the form $2^p - 1$ where $p$ is a prime,
A000668, $43\%$)
Sophie Germain primes ...
5
votes
1answer
216 views
Does this $\zeta(s)$ identity have a name?
I have generalized the product from this thread:
Let $s=2n+1$ for $n\ge1$,
$$\zeta (s)=\frac{\zeta (2 s)}{\zeta (2)} \prod _{n=1}^{\infty } \frac{\sum _{i=0}^{s-1}(-p_n){}^i}{(p_{n}-1)p_{n}^{s-2}}$$ ...
5
votes
1answer
374 views
Is it known or new? [duplicate]
Possible Duplicate:
Starting digits of 2^n
While I was randomly working with number patterns, I came along with some interesting pattern which seems to turn to a conjecture in fact.
My ...
1
vote
1answer
96 views
I noticed a pattern, does this have a name?
First of all I am a programmer, not a mathematician, so I may articulate what I am trying to say very poorly. I was working with powers of $2$ when I noticed a relationship I had never noticed before. ...
40
votes
1answer
1k views
Does $|n^2 \cos n|$ diverge to $+\infty$?
I was recently exposed to the problem of deciding whether
$$ \lim_{n \to +\infty} |n \cos n| = +\infty$$
where the limit is taken over the integers. As $|\cos n|$ oscillates throughout the interval ...
7
votes
3answers
712 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 ...
4
votes
1answer
132 views
Problems about consecutive semiprimes
I was playing around with semi-prime numbers and I made two conjectures, which are:
Given any integer $a$, at least one of $a,(a+1),(a+2)$ or $(a+3)$ is not semi-prime.
There are infinitely many ...
3
votes
3answers
491 views
Goldbach conjecture
I have seen the following in my observation. I am so sorry, if I am wrong. I do not know how far I am right. But, I feel that, it might be correct. If correct, Please let me know that way to proceed ...
1
vote
2answers
306 views
twin prime conjecture
Whether I am correct or wrong I don't know. If there are any corrections, please let me know.
Let $p_n$ = product of all primes. (of course we can go still beyond as we know $p_n$ is infinite). Now ...
15
votes
3answers
504 views
Interesting Property of Numbers in English
I was playing with the letters in numbers written in English and I found something quite funny. I found that if you count the number of letters in the number and write this as a number and then count ...
-2
votes
1answer
197 views
Number theory conjecture [closed]
Let us observe the following pattern
$N - p_1 = m_1, N - p_2 = m_2, \ldots , N - p_r = m_r$; take $p_1 = 3$ and $p_2 = 5,\ldots$ notice that $p_r$ is the larger prime less than or equal to square ...
7
votes
1answer
202 views
What is the importance of 3n in the Collatz Conjecture?
I'm not mathematician, so forgive me if I make wrong assumptions. I was wondering what the importance of the $3n$ is in the Collatz Conjuncture.
If you just do $n + 1$, it seems you'll end up at $1$ ...
1
vote
0answers
165 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
2answers
221 views
If $a$ is even and $b$ is odd then $\gcd(2^{a}+1,2^{b}+1)=1$
How to prove that:
If $a$ is even and $b$ is odd then $\gcd(2^{a}+1,2^{b}+1)=1$
This statement is generalization of the statement from my previous question.
I have checked for many $(a,b)$ ...
1
vote
2answers
233 views
If $\gcd(a,b)=1$ , and $a$ is even and $b$ is odd then $\gcd(2^{a}+1,2^{b}+1)=1$?
How to prove that:
$\gcd(a,b)=1 \Rightarrow \gcd(2^{a}+1,2^{b}+1)=1$ ,where $a$ is even and $b$ is odd natural number
For example:
$\gcd(2^8+1,2^{13}+1)=1 , \gcd(2^{64}+1,2^{73}+1)=1$
I know ...
1
vote
1answer
143 views
Infinitely many primes of the $\sum_{i=0}^{a} n^{i}+m\cdot\sum_{i=0}^{b} n^{i}$ form?
How to show that there is infinitely many prime numbers of the form:
$p=\sum_{i=0}^{a} n^{i}+m\cdot\sum_{i=0}^{b} n^{i}$
where: $m\in \mathbb{Z}^{*}$ , $a,b,n\in \mathbb{N}$ , $\gcd(a+1,b+1)=1$
For ...
11
votes
1answer
269 views
Is there any theoretical indication that this conjecture of Catalan could be true?
Belgian mathematician Catalan in $1876$ made next conjecture:
If we consider the following sequence of Mersenne prime numbers: $2^2-1=3 , 2^3-1=7 , 2^7-1=127 , 2^{127}-1$ then $$2^{2^{127}-1}-1$$ is ...
3
votes
2answers
367 views
A conjecture about the form of some prime numbers
Let $k$ be an odd number of the form $k=2p+1$ ,where $p$ denote any prime number, then it is true that for each number $k$ at least one of $6k-1$, $6k+1$ gives a prime number.
Can someone prove or ...
13
votes
2answers
390 views
open conjectures in real analysis targeting real valued functions of a single real variable
I am hoping that this question (if in acceptable form) be community wiki.
Are there any open conjectures in real analysis primarily targeting real valued functions of a single real variable ? (it may ...
12
votes
4answers
791 views
How to propose a conjecture
What are the basic things (about when and how) to be kept in mind while proposing a conjecture in Mathematics. Should it accompany solid efforts at proof. When should any one think of proposing a ...
9
votes
1answer
350 views
Continuous Collatz Conjecture
Has anyone studied the real function
$$ f(x) = \frac{ 2 + 7x - ( 2 + 5x )\cos{\pi x}}{4}$$ (and $f(f(x))$ and $f(f(f(x)))$ and so
on) with respect to the Collatz conjecture?
It does what Collatz ...
