The conjectures tag has no wiki summary.
42
votes
12answers
3k 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 ...
39
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 ...
17
votes
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:
$$ \zeta(s) = 0 ...
16
votes
1answer
357 views
Status of a conjecture about powers of 2
I recently saw a conjecture on a blog ( http://blog.tanyakhovanova.com/?p=311 ) which the author refers to as the 86 conjecture. The conjecture claims that all powers of 2 greater than $2^{86}$ have a ...
14
votes
3answers
664 views
The $5n+1$ Problem
The Collatz Conjecture is a famous conjecture in mathematics that has lasted for over 70 years. It goes as follows:
Define $f(n)$ to be as a function on the natural numbers by:
$f(n) = n/2$ if $n$ ...
1
vote
1answer
143 views
Infinitely many primes of the $\sum_{i=0}^{a} n^{i}+m\cdot\sum_{i=0}^{b} n^{i}$ form?
How to show that there is infinitely many prime numbers of the form:
$p=\sum_{i=0}^{a} n^{i}+m\cdot\sum_{i=0}^{b} n^{i}$
where: $m\in \mathbb{Z}^{*}$ , $a,b,n\in \mathbb{N}$ , $\gcd(a+1,b+1)=1$
For ...
1
vote
1answer
203 views
Converse of Collatz Conjecture
How to write a pseudocode program that halts only if the Collatz Conjecture is
false.
Thanks much in advance!!!
-2
votes
1answer
196 views
Number theory conjecture [closed]
Let us observe the following pattern
$N - p_1 = m_1, N - p_2 = m_2, \ldots , N - p_r = m_r$; take $p_1 = 3$ and $p_2 = 5,\ldots$ notice that $p_r$ is the larger prime less than or equal to square ...
14
votes
1answer
206 views
Density of odd numbers in a sequence relating base 2 and base 3 expansion
Define the function
$$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 ...
2
votes
2answers
220 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 $(a,b)$ ...
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
498 views
Why proof by induction fails for Goldbach's conjecture?
Can anyone clarify why induction method fails for this conjecture?
1
vote
2answers
301 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 ...
1
vote
2answers
231 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 ...
0
votes
1answer
285 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 ...
