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

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0
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4answers
102 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
200 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
157 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
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1answer
280 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
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0answers
89 views

Question regarding the status of Erdős' conjectures

Has Erdős' conjecture on arithmetic progressions or his conjecture on Sylvester's sequence been proven?
5
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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
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1answer
174 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
827 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
153 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
246 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
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3answers
115 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 ...
7
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0answers
259 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
168 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
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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
137 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
292 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
269 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
338 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
905 views

Why proof by induction fails for Goldbach's conjecture?

Can anyone clarify why induction method fails for this conjecture?
11
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1answer
376 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
301 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: ...
3
votes
2answers
607 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
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1answer
266 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
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1answer
409 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
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 ...
10
votes
4answers
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
210 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
4answers
741 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
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2answers
390 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
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3answers
585 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
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1answer
273 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
236 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 $1$ ...
2
votes
0answers
284 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
382 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
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2answers
310 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
166 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
288 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
410 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
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2answers
515 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
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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 ...
11
votes
1answer
399 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 ...
16
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1answer
475 views

Status of a conjecture about powers of 2

I recently saw a conjecture on a blog ( http://blog.tanyakhovanova.com/?p=311 ) which the author refers to as the 86 conjecture. The conjecture claims that all powers of 2 greater than $2^{86}$ have a ...
13
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3answers
1k views

The $5n+1$ Problem

The Collatz Conjecture is a famous conjecture in mathematics that has lasted for over 70 years. It goes as follows: Define $f(n)$ to be as a function on the natural numbers by: $f(n) = n/2$ if $n$ ...
4
votes
1answer
208 views

A conjecture of parallelogram inside convex and central symmetric curve

Assume Q is a convex central symmetric curve, whose area is $\displaystyle S$. The area of the maximum parallelogram inside Q is $\displaystyle S'$. How to prove the conjecture that $\displaystyle ...
22
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4answers
1k views

Proving a known zero of the Riemann Zeta has real part exactly 1/2

Much effort has been expended on a famous unsolved problem about the Riemann Zeta function $\zeta(s)$. Not surprisingly, it's called the Riemann hypothesis, which asserts: $$ \zeta(s) = 0 ...
8
votes
3answers
632 views

Branches of mathematics not having a general method to solve

I studied applied math, so each course (except abstract algebra) was dedicated to solution of a similar problems. After those courses it seems that every branch of mathematics has a developed theory ...
59
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12answers
5k views

Conjectures that have been disproved with extremely large counterexamples?

I just came back from my Number Theory course, and during the lecture there was mention of the Collatz Conjecture. I'm sure that everyone here is familiar with it; it describes an operation on a ...