1
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1answer
25 views

Random conjecture

For any numbers $a$ and $b$ so that $a>b$ and $\text{gcd}(a, b)=1$, there exists a $c\in\mathbb{N^+}$ so that $a+bc$ is prime. I've only just tested this out with a few numbers, but I'm curious as ...
0
votes
2answers
19 views

On extracting primes from coprimes

Proof or disprove the following statement - There exists infinitely many $a$ and $b$ which are pair of co-prime integers , either $ab+1$ or $ab-1$ is prime. Motivation- Looking at some twin prime ...
0
votes
1answer
135 views

For all $n$, $9^n + 25^n - 1$ has a prime factor with $7$ in its decimal representation?

Let $x_n$ be a sequence of positive integers defined by $x_n=9^n + 25^n -1$ for all $n \ge 2$ I conjectured that there exists at least one prime divisor of $x_n$ which contains $7 $ in its decimal ...
5
votes
0answers
99 views

Are $ut + 1$ and $ut + t + 1$ both prime for some t for any $u$?

Conjecture : For any natural number $u$, there is a natural number $t$ such that $ut + 1$ and $ut + t + 1$ are both prime. So we get a solution of the equation $$au - b(u+1) = -1$$ with prime ...
7
votes
1answer
158 views

Proving infinitude of primes in a certain form.

Here I have the following conjecture -Let $$S_1(n)= \frac{(n-1)! +1}{n}$$ then there exist infinite prime numbers $p$ for which $S_1(p)$ is prime. And I don't know how to prove it. EDIT Let ...
2
votes
2answers
79 views

An approach to Andrica's conjecture

Andrica's conjecture states that $\sqrt{p_{n+1}}-\sqrt{p_n} < 1$. but solving for $n=1,2,\dotsc$ yields n=1, $\sqrt{p_{2}}-\sqrt{p_1} < 1$=>$\sqrt{p_{2}}<\sqrt{p_1}+1$ n=2, ...
2
votes
0answers
99 views

Is this a satisfiable proof (hypothetical) -yet simple one- of Goldbach's conjecture? [closed]

Last night I found a paper by a guy named Miles MATHis that claims to simply prove Goldbach's conjecture. Abstract: Here I solve Goldbach's Conjecture by the simplest method possible. I do this by ...
1
vote
0answers
106 views

Proving there exists prime numbers between the squares of prime numbers

Conjecture: $\forall $ $p_{n}$, $p_{n+1} \in \mathbb{P}$, $\:$ $p^2_{n+1} = p^2_{n} +\omega_{n} p_{n} + \phi_{n} : \phi_{n} , \omega_{n} \in \mathbb{N} $ and $ \phi_{n} < p_{n}$, $\:$ ...
2
votes
2answers
113 views

Does there exist a prime number within the interval?

Conjecture $\forall p_{n}\in \mathbb{P} : n\geq3, \: \exists p_{m}\in \mathbb{P} : 3p_{n} - 4 \geq p_{m} > \sqrt{2(p^2_{n+1} - 1)} $ How would you go about proving/disproving this?
4
votes
1answer
149 views

A conjecture about the difference between consecutive primes with respect to a prime number squared.

Conjecture If we have two consecutive prime numbers $p_{a}$ and $p_{a+1}$, and two other consecutive primes $p_n$ and $p_{n+1}$, so that $p_{a} < p_{a+1} < p^2_{n+1}$, then $p_{a+1} - ...
1
vote
1answer
117 views

The name and proof of a conjecture on prime intervals

Conjecture: There exists at least one prime number $p_{m}$ : $ap_{n} < p_{m} < (a+1)p_{n}$, $\forall$ $a \in \mathbb{N}$ and $\forall$ $p_{n}$ $\in \mathbb{P} $ if $(a+1)p_{n} < ...
0
votes
1answer
605 views

Checking the Harald Helfgott proof of the little Goldbach conjecture without a public release of numerical checks?

A few month ago, a proof of the little/ternary Goldbach conjecture has been claimed by Harald Helfgott with three articles: Major arcs for Goldbach's theorem Minor arcs for Goldbach's theorem ...
3
votes
1answer
150 views

Weaker Version of “Goldbach's Other Conjecture”

Taken from problem 46 on Project Euler: It was proposed by Christian Goldbach that every odd composite number can be written as the sum of a prime and twice a square. $9 = 7 + 2 \times 1^2$ ...
2
votes
0answers
155 views

Approximation to $\pi(x)$ conjecture.

A friend conjectured that $\left[\prod_{k=1}^{a_j <\sqrt{x}} \left(1-\frac{1}{a_k}\right)\right] x$ is usually closer to $\pi(x)$ than $\operatorname{Li}(x)$ is for some (fixed) sequence of ...
1
vote
2answers
183 views

“Goldbach's other conjecture” and Project Euler - writing 1 as a sum of a prime and twice a square

From Problem 46 of Project Euler : It was proposed by Christian Goldbach that every odd composite number can be written as the sum of a prime and twice a square. $$9 = 7 + 2 \cdot ...
5
votes
1answer
279 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
177 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
80 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
144 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
101 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
155 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
771 views

Why proof by induction fails for Goldbach's conjecture?

Can anyone clarify why induction method fails for this conjecture?
11
votes
1answer
331 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
549 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
255 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
195 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
254 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
160 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
282 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
404 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 ...