# Tagged Questions

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### Intuitively, what separates Mersenne primes from Fermat primes?

A Mersenne prime is a prime of the form $2^n-1$. A Fermat prime is a prime of the form $2^n+1$. Despite the two being superficially very similar, it is conjectured that there are infinitely many ...
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### Find all $n$ so that $c_n$ $>$ $\pi(n^2)$

Find all $n$ $\in$ $\mathbb{N}$ so that $p_{c_n}$ $>$ $n^2$ where $p_n$ denotes the $n$-th prime and $c_n$, the $n$-th composite. I have tried doing the problem using The stronger version of ...
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### Is this a conjecture or an already existing one??

Does the following inequality hold? $p(n)\leq 2^n,$ where $p(n)$ is the $n$th prime. If this is true then it follows that: If $p(n)=p(m)^x+p(o)^y$, then $\max[x,y] \le n$.
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### Zhang's theorem and Polignac's conjecture

Yitang Zhang made a groundbreaking discovery when he proved that there are infinitely many pairs of prime numbers which differ by less than $70,000,000$. Zhang's theorem has been significantly ...
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### Pi and the sum of reciprocals of primes?

So I know that $$\sum_{\underset{\Large p\; prime}{p=1}}^{\infty}\frac{1}{p}$$ blows up. But doing some fun on mathematica I found out that when the sum isn't infinite, it was so close to $3$ and I ...
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### Prove that there exists an $m$ such that for any $n>m$ there exists at least one prime between $c_n$ and $n$

Let $c_n$ be the $n$-th composite. Then the problem is to prove that- $\pi(c_n)-\pi(n)>0$ $\forall n>m$ I have tried to progress in the problem using an elementary approach. So far I have ...
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### How to prove that every $l$ (such that $2 \leq l \leq \lfloor \sqrt{k^2+2n+1} \rfloor$) divides at least one of the following numbers?

$k^2+2n, k^2, k^2+1, 2n, 2n+1$, (for some $n$) if $k$ is even and $0 < n < k$. I have no idea of how to prove that. I'm working on Legendre's conjecture. Update 1: Yes, for $n=0$ all $l$ ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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, ...
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### 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 ...
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### 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 $\phi_{n} < p_{n}$, $\:$ ...
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### 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?
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### 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 ...
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### 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. $9 = 7 + 2 \times 1^2$ ...
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### 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 ...
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. $$9 = 7 + 2 \cdot ... 1answer 293 views ### Is this number theory conjecture known to be true? I've been working on proving that there is always a prime between n and 2n, and also that there is always a prime between n^2 and (n+1)^2 (Legendre's conjecture). I believe I've proven those ... 2answers 195 views ### Finding a counterexample to a Prime Factorization Conjecture Let \mathbb{Z}_{\geq 2} be the set of natural numbers starting at 2:$$\mathbb{Z}_{\geq 2}= \{2, 3, 4, 5,\ldots\}.$$An natural number's prime factorization is odd if the total number of primes in ... 0answers 81 views ### Prime clasfication by some constructive function How to prove or justify the following:$$ 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), $$The above statment can ... 1answer 152 views ### Goldbach conjecture and primes I need some clarification on (1) Is there any proof to say Mersenne primes M_p are finite or infinite? if there, could you share here.. (2) If Goldbach is conjecture is true, how you can justify the ... 3answers 113 views ### Primes + Inetvel + conjecture on primes a) Can we establish a proof, there exists infinitely many primes of the form n^2 + 1. Why is the unit digit of such a prime p always 1 or 7? Is there any reasonable procedure or concept for the fact ... 1answer 165 views ### How to prove the equivalence between the two statements of ABC conjecture? The ABC conjecture stated by wikipedia says the following statements are equivalent: I. For \epsilon>0, there are finite coprime triple (a,b,c) satisfying a+b=c such that ... 1answer 850 views ### Why proof by induction fails for Goldbach's conjecture? Can anyone clarify why induction method fails for this conjecture? 1answer 367 views ### The Goldbach Conjecture and Hardy-Littlewood Asymptotic A source I am reading refers to the Goldbach conjecture (that every even number is the sum of two primes), and then immediately follows with the "Hardy-Littlewood conjecture" that \sum ... 2answers 594 views ### Disprove the Twin Prime Conjecture for Exotic Primes The List of unsolved problems in mathematics contains varies conjectures of exotic primes like: Mersenne primes (of the form 2^p - 1 where p is a prime, A000668, 43\%) Sophie Germain primes ... 1answer 265 views ### Does this \zeta(s) identity have a name? I have generalized the product from this thread: Let s=2n+1 for n\ge1,$$\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}}$$... 1answer 208 views ### Problems about consecutive semiprimes I was playing around with semi-prime numbers and I made two conjectures, which are: Given any integer a, at least one of a,(a+1),(a+2) or (a+3) is not semi-prime. There are infinitely many ... 0answers 280 views ### Can the openness of Gilbreath's conjecture be reduced to proof of the Riemann hypothesis? As an information theoretician, it's a personal hobby of mine to find elegant analogies to open mathematical problems. After all, they have a profound impact on how I conduct my research and how I ... 1answer 165 views ### Infinitely many primes of the \sum_{i=0}^{a} n^{i}+m\cdot\sum_{i=0}^{b} n^{i} form? How to show that there is infinitely many prime numbers of the form: p=\sum_{i=0}^{a} n^{i}+m\cdot\sum_{i=0}^{b} n^{i} where: m\in \mathbb{Z}^{*} , a,b,n\in \mathbb{N} , \gcd(a+1,b+1)=1 For ... 1answer 286 views ### Is there any theoretical indication that this conjecture of Catalan could be true? Belgian mathematician Catalan in 1876 made next conjecture: If we consider the following sequence of Mersenne prime numbers: 2^2-1=3 , 2^3-1=7 , 2^7-1=127 , 2^{127}-1 then$$2^{2^{127}-1}-1 is ...
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 ...