Prime numbers are natural numbers greater than 1 not divisible by any smaller number other than 1. This tag is intended for questions about, related to, or involving prime numbers.

learn more… | top users | synonyms

8
votes
1answer
442 views

Conjecture about primes and the factorial: for all primes $p>5$, must there exist a prime $q<p$ such that $q\equiv m!\pmod p$ for some $2<m<p$?

Below $0\notin\mathbb N$. Further corrected conjecture: For all prime numbers $p>5$ there exist a prime number $q<p$ such that $q\equiv m!\!\pmod p$, $2<m<p$. or Given a prime ...
8
votes
1answer
225 views

Are there infinitely many primes $p$ such that $\frac{(p-1)! +1}{p}$ is prime?

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 $...
9
votes
1answer
147 views

Does any sum of twin primes, where the sum is greater than 12, also represents the sum of 2 other distinct primes?

I was in the midst of proving a conjecture when I came across an observation that led me to forming a potentially new conjecture. The conjecture goes as follows: Any given sum of twin primes (...
1
vote
1answer
53 views

Prime conjecture containing primorial: the difference between the primorial $n\#$ and the smallest prime $p > n\# + 1$ is always a prime

Help me find the exact conjecture statement. What I roughly remember is that it stated that the difference between primorial $n\#$ (product of first $n$ primes) and "some" larger number than the ...
8
votes
2answers
300 views

A conjecture about the prime function $p_n$: $p_m \cdot p_n >p_{m \cdot n}$

While testing my system Zet for computational mathematics I find possible relations now and then. The latest is: Conjecture: For all $(m,n)\in\mathbb Z_+^2$ except $(3,4),(4,3) \text{ and } (4,4)$...
4
votes
1answer
134 views

Is this a known conjecture? Given odd primes $p,q$ with $p + q$ sufficiently large, must there exist a different pair $p',q'$ with $p+q = p'+q'$?

Conjecture: There is a natural number $N\in\mathbb N$ such that given odd primes $p,q$ with $p+q>N$ there are primes $p',q'$ where $p' \notin \{p,q\}$ such that $p+q=p'+q'$. Is this known?
1
vote
1answer
66 views

For every prime $p > 3$ that is $3$ mod $4$, does $q+1 \mid p-q$ for some other prime $q$?

Yet another random conjecture about primes: Given a prime $p>3$ of the form $4n+3$. Then there exist a prime $q<p$ such that $q+1\mid p-q$. Verified for all $p<100000$.
6
votes
0answers
67 views

Does every power of two arise as the difference of two primes?

Conjecture: For each $n\in\mathbb N$ there are primes $q<p$ with $p-q=2^n$. Verified for $n\leq 26$: ...
2
votes
0answers
139 views

A conjecture about primes: if $a,b$ are coprime and not both odd, is $A(a,b,m)$ finite for some $m$?

Let $p_n$ be the $nth$ prime and define $p_n^{(m)}$ by $p_n^{(1)}=p_n$ and $p_n^{(m+1)}=p_{p_n^{(m)}}$: $p_n^{(2)}=p_{p_n}$, $\;p_n^{(3)}=p_{p_{p_n}}$ and so far... For some coprime numbers $a,b$, ...
25
votes
4answers
613 views

Does this conjecture about prime numbers exist? If $n$ is a prime, then there is exist at least one prime between $n^2$ and $n^2+n$.

I made an observation on prime numbers, want to check if any conjecture already exist or not? I am a computer programmer by profession and I am interested in number theory. As like many others I am ...
4
votes
3answers
242 views

Does the sequence $q(n)=3n+1+\frac{1-(-1)^n}{2}$ generate all the prime numbers?

Define $$q(n)=3n+1+\frac{1-(-1)^n}{2} \quad, \quad n\in \mathbb N.$$ $$1,5,7,11,13,17,19,23,25,29,31,35,\dots$$ It seems like this formula gives all primes $>3$ (although not just primes of ...
4
votes
3answers
44 views

The sum of more than two consecutive natural numbers cannot be prime.

The sum of more than two consecutive natural numbers cannot be prime. Is the statement true and is there any way to prove it? I was able to prove that the sum of an odd amount of consecutive ...
1
vote
0answers
22 views

Asymptotic Growth of Function of Prime Counting Function

Consider $f(x)$ defined by $$f(x)=\sum_{k=1}^\infty \pi\Big{(}\frac{x}{k}\Big{)}$$ How may one another function $g(x)$ be defined such that $$\lim_{x\to\infty}\frac{f(x)}{g(x)}=1$$I have tried $g(x)=c\...
1
vote
1answer
130 views

Discrete log for prime powers

I was fiddling around and found that a function of the form $$L_b (x)=\left(\frac{b^{\phi (p^k)}-1}{p^k}\right)^{-1}\left(\frac{x^{\phi (p^k)}-1}{p^k}\right) \mod p^k$$ seems to behave similarly to a ...
3
votes
3answers
111 views

The greatest common divisor of multiple numbers

What is the cardinality of the following set $\{{\bf x}=(x_1,\ldots,x_d): \text{each } x_i\in \{ 1,\dots,n \},\text{ and } \gcd({\bf x})=1\}$, where $\gcd({\bf x})$ is the greatest common divisor of ...
3
votes
2answers
30 views

For which $0\leq a<p^2$, where $p$ is an odd prime, we have that $(2p-1)!\equiv a\mod{p^2}$

Let $p$ be an odd prime. I need to find for which $0\leq a < p^2$, $(2p-1)!\equiv a\mod{p^2}$. If $a\equiv (2p-1)!\mod{p^2}$, then we have that $a = kp^2 + (2p-1)!$, and therefore $p\mid a$, ...
3
votes
1answer
54 views

Prime Number Theorem and the Riemann Zeta Function

Let $$\zeta(s) = \sum_{n=1}^\infty \frac{1}{n^s}$$ be the Riemann zeta function. The fact that we can analytically extend this to all of $\mathbb{C}$ and can find a zero free region to the left of the ...
3
votes
3answers
41 views

Show that for $x,y,z\in\mathbb{Z}$, if $x$ and $y$ are coprime, then $\exists n\in\mathbb{Z}$ such that $z$ and $y+xn$ are coprime.

Not sure where to start on this one. I understand that coprime indicates that their GCD is 1, but I am somewhat confused how to proceed.
3
votes
1answer
74 views

Is one of $k+1^2,$ $k+2^2,$ …, $k+N^2$ always prime?

I know that the Bunyakovsky conjecture is still open, so we can't prove that there exist primes of the form $n^2+k$ for a given $k$. But suppose that they do: is the least $n$ such that $n^2+k$ is ...
2
votes
1answer
62 views

If $q\mid 2^p + 3^p$ then $q \gt p$

Let $p, q$ positive prime numbers, $q > 5$. Prove that if $q \mid \left(2^{p} + 3^{p}\right)$ then $q > p$. First, it's clear that $p \ne q$ because, using Fermat's little theorem, $2^p = ...
2
votes
2answers
45 views

An unexplained condition on $a$ in a proof on the primes?

Lemma A positive integer $n$ is a prime if $(n,p) = 1$ for every prime integer $p \leq \sqrt{n}$ Proof in my text Let $(n,p) = 1$ for every prime $p \leq \sqrt{n} \:$. Suppose $n$ is not a ...
5
votes
0answers
78 views

Is the error I noticed a harmless typo?

Here http://arxiv.org/PS_cache/arxiv/pdf/1002/1002.0442v1.pdf , at page $2$ at the bottom, it is stated that the number of primes not exceeding $x$, denoted by $\pi(x)$, satisfies the double-...
3
votes
1answer
169 views

Generalization of Inkeri's primality test

How to prove that following hypothesis is true ? Definition Let $P_m(x)=2^{-m}\cdot \left(\left(x-\sqrt{x^2-4}\right)^{m}+\left(x+\sqrt{x^2-4}\right)^{m}\right)$ , where $m$ and $x$ are ...
2
votes
0answers
139 views

Are all theorems usable? [closed]

The (revised) question to answer: Can anyone give an example of a serious proof using this funny (revised) theorem? For any natural number $n$ and prime $p<n-1$ there exist a prime $q$ ...
1
vote
1answer
46 views

Simpler way to compress this exponentiation?

I am trying to find when the following is true: Let $H =(10k)^b \bmod 6(p-1)$ Let $J = 10^{H} \bmod 9p$ For some prime $p > 5$ and large $k,b$. I am trying to find when $J$ is equal to $1$. ...
2
votes
1answer
39 views

Statement regarding primes $ \le n$

Following is the statement I believe is true, but can't prove. Let $n$ be a natural. Let the primes less than equal to $\sqrt{n}$ be $p_1,p_2,...,p_k$. Let $\alpha_i$ be the greatest natural ...
6
votes
0answers
61 views

The Greatest Common Divisor of All Numbers of the Form $n^a-n^b$

For fixed nonnegative integers $a$ and $b$ such that $a>b$, let $$g(a,b):=\underset{n\in\mathbb{Z}}{\gcd}\,\left(n^a-n^b\right)\,.$$ Here, $0^0$ is defined to be $1$. (Technically, we can also ...
1
vote
1answer
48 views

Help on an application of Dirichlet's theorem for primes in progression

Suppose that I have an infinite sequence of positive integers $$a_1,\ldots,a_m,\ldots$$ with the following recursion $$a_{m+1} -a_m =b(m+1)$$ So that $$a_{m+1} =b(m+1) +a_m$$ Suppose ...
9
votes
3answers
262 views

On the difference between consecutive primes

Let $(p_n)$ be the sequence of prime numbers and $g_n = p_{n+1} - p_n$ Question: Is it known that $g_n \le n$? Remark: it's known that $g_n < p_n^{\theta}$ with $\theta = 0.525$ for $n$ ...
0
votes
0answers
18 views

Number of prime exponents for Generalized Mersenne Primes

Please help me on the following scenario(s): Estimate the number of primes $p$ less than or equal to $x$ such that there is a prime of the form ${(a+1)}^p$ $-$ $a^p$ for all $a$ < $50$? What is ...
1
vote
2answers
43 views

is the multiplication of n consecutive prime numbers starting with 2 plus 1 prime?

The question kinda tell everything for itself, let P(n) be the n-th prime number, is $(\Pi_1^n P_n)+1$ prime ?
0
votes
0answers
28 views

On Dirichlet series and Firoozbakht's conjecture

On assumption of the Firoozbakht's conjecture (this is the Wikipedia, but the reference is for Carlos Rivera's Page) one has that can writes informally the Dirichlet series in LHS of this inequality $$...
0
votes
0answers
48 views

Is a tight concrete bound for the error-term in the prime-number-theorem known?

Here : https://en.wikipedia.org/wiki/Prime_number_theorem it is mentioned that $$\pi(x)=Li(x)+O(xe^{-a\sqrt{ln(x)}})$$ What is a tight upper bound for $|\pi(x)-Li(x)|$ in concrete terms ? The ...
3
votes
1answer
68 views

Does a sequence based on hereditary factorisation always terminate?

The well-known Goodstein sequences are based on the hereditary base-$b$ notation, where you don't just present the digits in base $b$, but also the corresponding exponents etc. That lead me to the ...
3
votes
2answers
112 views

characterisation of $n$ as prime using min values of $x$ such that $nx+1$ or $nx$ is square

Let $n\ge 5$ be an odd integer and $k\ =\ \min\{x\in\mathbb{N}\colon nx+1\text{ is a perfect square}\}$ $l\ =\ \min\{x\in\mathbb{N}\colon nx\text{ is a perfect square}\}$ Prove that $n$ is a prime ...
1
vote
0answers
48 views

a consequence of Prime Number Theorem

By Prime Number Theorem we have $\lim_{n\to\infty}\frac{p_{n+1}}{p_n}=1$, so $\frac{p_{n+1}}{p_n}=1+a_n$ where $a_n\to 0$. How fast does $(a_n)$ converge to $0$ ? Does for example $a_n\ln n$ or $...
3
votes
1answer
62 views

Longest sequence of primes where each term is obtained by appending a new digit to the previous term

What is the longest known sequence of primes where each new term is obtained by appending a new decimal digit to the previous term? Examples: $$(2,23,233,2333,23333)$$ There are no more members in ...
3
votes
0answers
98 views

How to test if $n!+1$ is prime or not?

for $n=0,1,2,3,11,27,37,41,73,77,116,154,320,340,399,427,872,1477,6380,26951,...$ $$n!+1$$ is prime. But how can you proof (with 100% certantiy) thats the case? Especially for the larger ones. For ...
-1
votes
1answer
81 views

Is there any formula which gives better approximation than this formula?

Let $g(n),f(n)$ be functions of $n \in \mathbb{N}$. $g(n)=(n−1)^\frac{1}{n−1}$ $f(n)=\frac{a(g(n)^n)+(g(n)+(\frac{b}{n}))^n}{2}$ $P(n)=(f(n))\log_e (f(n))$ $P(n)$ gives the $n$th Prime Number. $a=...
2
votes
2answers
103 views

Asymptotic expression for sum of first n prime numbers?

Is one known? If not, what are the best known bounds? Is there reason to think that an asymptotic expression is beyond current methods if none exists?
15
votes
3answers
1k views

A condition for being a prime: $\;\forall m,n\in\mathbb Z^+\!:\,p=m+n\implies \gcd(m,n)=1$

If $\;p=m+n$ where $p\in\mathbb P$, then $m,n$ are coprime, of course. But what about the converse? Conjecture: $p$ is prime if $\;\forall m,n\in\mathbb Z^+\!:\,p=m+n\implies \gcd(m,n)=1$ ...
1
vote
1answer
25 views

Anti-associativity of product of sum of squares

$\newcommand{\P}{\mathbb{P}}$$\newcommand{\Z}{\mathbb{Z}}$ Let $\P$ be the set of prime numbers congruent to $1 \pmod 4$. I know that for every $p \in \P$ there's a unique couple $(a^2,b^2)\in \Z^2$ ...
2
votes
0answers
45 views

Heegner Prime visualizations

The Heegner numbers are 1, 2, 3, 7, 11, 19, 43, 67, 163. The ring of integers $\textbf{Q}(\sqrt{-d})$ have unique factorizations. 1 gives the Gaussian integers. 3 gives the Eisenstein integers. 7 ...
31
votes
4answers
1k views

Can a Mersenne number ever be a Carmichael number?

Can a Mersenne number ever be a Carmichael number? More specifically, can a composite number $m$ of the form $2^n-1$ ever pass the test: $a^{m-1} \equiv 1 \mod m$ for all intergers $a >1$ (Fermat'...
0
votes
3answers
54 views

No primes in $f(n) = n(n+3)/2$ except 2 and 5

How can I prove that the sequence $f(n) = n(n+3)/2 = 0, 2, 5, 9, 14, 20, 27, 35, 44, ...$ does not contain primes except $2$ and $5$?
-2
votes
1answer
52 views

Is this possible to solve? [closed]

If a, b, c are different prime numbers such that (a-b)(a-c) = 255, find the value of b + c.
3
votes
2answers
210 views

Percentage of Composite Odd Numbers Divisible by 3

What is the percentage of odd composite positive numbers divisible by 3? In that same vein, what is the percentage of odd composite positive numbers divisible by 5? And, for the future, what is the ...
5
votes
3answers
64 views

Does the PNT establish a connection between primes and the logarithm?

The prime number theorem states that $$\pi(x) \sim \frac x {\ln(x)}$$ Morally, this seems to suggest that there is a fundamental connection between primes and the natural logarithm. But since we're ...
2
votes
1answer
29 views

How to count the number of perfect square greater than $N$ and less than $N^2$ that are relatively prime to $N$?

I know a little about Euler's totient function that counts integer less than $N$ that are relatively prime to $N$. But I don't know how to modify the function for perfect square numbers, or maybe ...
9
votes
0answers
91 views

For any $x\in \mathbb{N}$ does there exist $m\in \mathbb{N}$ such that $2x+1+2m, 2x+1+4m$ are both prime?

Could someone please give me a proof (or counter example) for this (I believe it is true): For any $x$ (Whole Number) there exists some $m$ (Also Whole) such that $2x+1+2m$ and $2x+1+4m$ are both ...