Tagged Questions
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
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 ...
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 ...
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 ...
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
500 views
Why proof by induction fails for Goldbach's conjecture?
Can anyone clarify why induction method fails for this conjecture?
11
votes
1answer
232 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 ...
0
votes
1answer
292 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
217 views
Does this $\zeta(s)$ identity have a name?
I have generalized the product from this thread:
Let $s=2n+1$ for $n\ge1$,
$$\zeta (s)=\frac{\zeta (2 s)}{\zeta (2)} \prod _{n=1}^{\infty } \frac{\sum _{i=0}^{s-1}(-p_n){}^i}{(p_{n}-1)p_{n}^{s-2}}$$ ...
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 ...
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 ...
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
368 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 ...
