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

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18
votes
2answers
475 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 ...
18
votes
2answers
305 views

How to prove $4\times{_2F_1}(-1/4,3/4;7/4;(2-\sqrt3)/4)-{_2F_1}(3/4,3/4;7/4;(2-\sqrt3)/4)\stackrel?=\frac{3\sqrt[4]{2+\sqrt3}}{\sqrt2}$

I have the following conjecture, which is supported by numerical calculations up to at least $10^5$ decimal digits: ...
3
votes
1answer
72 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 = ...
4
votes
1answer
374 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
303 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
79 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 ...
16
votes
1answer
563 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 ...
1
vote
4answers
110 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
216 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
159 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
281 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
90 views

Question regarding the status of Erdős' conjectures

Has Erdős' conjecture on arithmetic progressions or his conjecture on Sylvester's sequence been proven?
5
votes
2answers
2k 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
175 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
911 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
82 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
161 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
316 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
119 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 ...
8
votes
0answers
274 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
1answer
174 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
59 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
145 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
294 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 ...
6
votes
2answers
281 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
355 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 ...
0
votes
1answer
1k views

Why proof by induction fails for Goldbach's conjecture?

Can anyone clarify why induction method fails for this conjecture?
11
votes
1answer
415 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 ...
3
votes
2answers
324 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: ...
4
votes
2answers
645 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
278 views

Does this $\zeta(s)$ identity have a name?

I have generalized the product from this thread: Let $s=2n+1$ for $n\ge1$, $$\zeta (s)=\frac{\zeta (2 s)}{\zeta (2)} \prod _{n=1}^{\infty } \frac{\sum _{i=0}^{s-1}(-p_n){}^i}{(p_{n}-1)p_{n}^{s-2}}$$ ...
6
votes
1answer
413 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
98 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. ...
44
votes
1answer
2k 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 ...
12
votes
5answers
1k 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
222 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
771 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
398 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 ...
16
votes
3answers
588 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 ...
-1
votes
1answer
278 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
245 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 $1$ ...
3
votes
0answers
312 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
409 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
312 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 ...
2
votes
1answer
167 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
293 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
418 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 ...
14
votes
2answers
527 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 ...
13
votes
4answers
2k 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 ...