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

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Is there a known mathematical equation to find the nth prime?

I've solved for it making a computer program, but was wondering there was a mathematical equation that you could use to solve for the nth prime?
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2answers
33 views

Arithmetic progression of primes question

Is it known whether for all positive integers $k$ there is an integer $a$ such that $a+30n$ is a prime number for all $n\in \{1,\ldots,k\}$?
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1answer
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In $1 < k < n-10^6$, what is $k$? (details in question)

This is a homework question of mine, I am not searching for the solution but rather what it means. It seems pretty straight forward but I am a little confused as to what the $k$ in $1 < k < ...
2
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1answer
51 views

Difference between sum of first n primes and prime(prime(n))

The seq is: -1, 0, -1, 0, -3, 0, -1, 10, 17, 20, 33, 40, 59, 90, 117, 140, 163, 218, 237, ... http://oeis.org/A239731 Is there's a formula looks like $$a(n) =n^2logn/2$$ for this seq?
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2answers
17 views

How does one compute how big the cycle of modding by a prime number is?

If I take the $k \in \mathbb{N}$ power of 10 and mod it by a large prime, I notice that the remainders repeat at some point. For instance $10^k mod~7$ seems to repeat every $8$th value of $k$. Given ...
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3answers
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First 10-digit prime in consecutive digits of e

Problem. What is the first 10-digit prime in consecutive digits of e. For those of you who don't know, in 2004 the answer produced a URL to a Google employment page (sort of). I just found about this ...
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2answers
43 views

Number of primes in $[30! + 2, 30! + 30]$

How to find number of primes numbers $\pi(x)$ in $[30! + 2$ , $30! + 30]$, where $n!$ is defined as: $$n!= n(n-1)(n-2)\cdots3\times2\times1$$ Using Fermat's Theorem: $130=1\mod31$, (since $31 \in ...
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1answer
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Need help in understanding $ord_p{a}$ as used in Theorem 1.1 from “On Some Exponential Equations Of S. S. Pillai”

I have a question about very early argument in the proof of Thereom 1.1. Theorem 1.1 of On Some Exponential Equations of S.S. Pillai states that if $a,b,c$ are nonzero integers with $a,b \ge 2$, then ...
0
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1answer
23 views

Truncatable primes

Why only 11 ? The number 3797 has an interesting property. Being prime itself, it is possible to continuously remove digits from left to right, and remain prime at each stage: 3797, 797, 97, and 7. ...
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1answer
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Questions about the proof that every odd integer is the sum of 5 primes

In http://arxiv.org/pdf/1201.6656.pdf, Tao proved that all odd numbers greater than 1 are the sum of 1, 3, or 5 primes. In page 36-37, he uses the fact that for all $x > 1.1\times10^{10}$, there ...
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1answer
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Solve for $p^a + 1 = 2\cdot q^b$ where $p,q$ are odd primes and $a,b \ge 2$

Now, clearly, $7^2 + 1 = 2\cdot5^2$. Is this the only solution? How would I prove this? Or if it is not the only solution, what would be the method to find other solutions? I'm not clear on how to ...
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0answers
36 views

Primes probability for $2^{2(ak+b)}-3$

I'm working on the following problem: If $x$ is a prime and of the form $ak+b$, is there a possibility to check, whenever $2^{2x}-3$ could be a prime or not, without calculating it or extracting ...
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3answers
50 views

primes like sophie germain primes

Show that if there exist infinitely many positive integers that cannot be represented as $xy+yz+zx$ for any natural $x,y,z$, then there exist infinitely many primes $p$ such that $2p-3$ is also prime. ...
0
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1answer
33 views

Proof of convergence of $L'\left(1,\chi\right)$

can someone give me a good reference for a clear proof of the convergence of $L'\left(1,\chi\right)$, $\chi$ real-valued, non-principal Dirichlet character? Thanks in advance.
1
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1answer
58 views

Does a solution exist where $p,q$ are odd primes and $p^a - q^b = p^c - q^d$ where $a > c > 1$ and $b > d > 1$

From my thinking so far, there is no solution. Is this an open question or is the answer well known? Here's my reasoning about this issue: If a solution exists, then: $$p^c(p^{a-c} - 1) = ...
4
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3answers
64 views

Connections between prime numbers and geometry

This might be a little open-ended, but I was wondering: are there any natural connections between geometry and the prime numbers? Put differently, are there any specific topics in either field which ...
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2answers
185 views

Distribution of prime numbers. Can one find all prime numbers?

I want to know if it is possible to find a formula that gives all the prime numbers? or can one find the distribution of prime numbers? I know that there is a set of ongoing research on prime ...
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0answers
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If $p^a \equiv -1 \pmod {q^b}$, is there anything that we can say about $a$ if $p,q$ are odd primes and $a,b > 1$

If $p^a \equiv 1 \pmod {q^b}$, then, from Carmichael's Theorem, we know that: $a = u\varphi(q^b) = u(q-1)(q^{b-1})$ where $u \ge 1$ Can we say anything similar if $p^a \equiv -1 \pmod {q^b}$
3
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2answers
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Proving that if $a,b > 1$, then $5^a - 3^b=16$ has only one solution with $a=2$ and $b=2$

This may be one of those problems that is easy to state but very hard to prove. I don't know. I have tried to show that there is only one solution but I have not made much progress. Here's what I ...
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1answer
46 views

Subset of prime numbers

Given a subset of prime numbers say $A$. It is given that for $p,q\in A$ we also must have $(pq+4)\in A$ . Show that $A=\phi$ My work so far: It is obvious that $2,3\notin A$ . because all the ...
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1answer
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Common Primes…

In SageMath, the software, I was trying to create a visualization of how common it is for a number to be prime. Can anyone help me with the code? I am a super beginner and lost. I was going to post ...
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3answers
915 views

How can I list all numbers relatively prime to X? (but less than X)

Given a number X how can I find all (or most) numbers that are relatively prime to and less than X? Ideally I'd like this function to tell me the largest primes first.
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2answers
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What value does the sum of the square of the reciprocals of the prime counting function converge to?

Using my only mathematical tool Wolfram Alpha, I noticed that $\sum_{n=2}^\infty \frac{1}{\pi(x)}$ seems to diverge. Naturally, the entered the following sum and, just like the zeta function, saw that ...
0
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0answers
29 views

Find $a,b,c \ge 2$ and $p,q$ odd primes where $p^a - 1 = c*q^b$

I've been recently thinking about finding primes $p,q$ where the power of one divides the power of the other when subtracted by $1$. For example, if we remove the requirement that $p,q$ be odd ...
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1answer
58 views

What is the 5000th happy prime number?

Im writing a program that finds the Nth happy prime number. I think it works, but to double check I want to compare what it returns for the 5000th happy prime number. The problem is, I dont know where ...
0
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0answers
23 views

Miller-Rabin primality test and testing one

I'm learning about Miller-Rabin primality test but in all the problems I see in the notes of a person I got them from, I see that even if he expressed the number as $2^1 \cdot something$, he still ...
6
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2answers
286 views

For all $n>2$: there exists $p$ prime: $n<p<n!$

The question is: For all $n>2$, where $n \in \mathbb Z$: there exists $p$ prime such that $n<p<n!$ Here is my Proof: $\forall$ $p<n: p|n!$, or $p$ divides $n!$ Since $n!$ and $n!-1$ ...
3
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1answer
61 views

On the sum of prime powers

Has anybody investigated the asymptotic growth rate of functions in the form of $$f(z,n)=\sum\limits_{p\le n}p^z$$ For $Re(z)\ge -1$. Of course $f(0,n)=\pi (n)$ has an ocean of research surrounding ...
0
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1answer
53 views

Is the product of two primes ALWAYS a semiprime?

I know by definition, a semi-prime has factors that are prime numbers. But what I'm unsure of, is if there is ever a case where the product of two prime numbers results in number with factors OTHER ...
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1answer
31 views

A function about prime numbers

Is there a function defined like this? $p(x)=1$ if $x$ is a prime, $p(x)=0$ if $x$ isn't a prime. If there is, what is the symbol of it?
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2answers
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Proof for the existence of primes not equal to $ap_\alpha +bp_\beta$ etc?

Is there a general proof to show that there exists prime numbers larger than $min(p_\alpha,p_\beta)$that are not equal to $ap_\alpha +bp_\beta$, given $p_\alpha,p_\beta\in\mathbb{P}-\left\{2\right\}$ ...
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0answers
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My first proof employing strong induction / complete induction (very simple number theory). Please mark/grade.

What do you think about my first proof employing strong induction? What mark/grade would you give me? Theorem Every natural number greater than 1 is a product of one or more primes. Proof First, ...
2
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1answer
27 views

Inequiality on prime gaps

Is there a good inequality for prime gaps. Like $p_{k}-p_{k-1}\leq f(k)$ ? In other words is there a known upper bound for $p_{k}-p_{k-1}$?
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Formula for the factorial triangle [closed]

Prime factorize factorials (below) $2!=2^1$ $3!=2^1 \cdot 3^1$ $4!=2^3 \cdot 3^1$ $5!=2^3 \cdot 3^1 \cdot 5^1$ $6!=2^4 \cdot 3^2 \cdot 5^1$ $7!=2^4 \cdot 3^2 \cdot 5^1 \cdot 7^1$ and construct ...
4
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3answers
275 views

Twin primes satisfy the congruence?

I need a justification for my observation. In general, we can list twin prime pairs in $(6n-1, 6n+1)$, where $n$ is some positive number. Of course, this is valid except $(3, 5)$. Now, I construct, ...
2
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1answer
55 views

Sum of reciprocals of every nth prime

I'm looking for a proof that $\displaystyle\sum_{n\mathop=1}^{\infty}\frac{1}{p_{kn}}$ diverges, where $p_n$ denotes the $n$th prime number and $k$ is a natural number. I know the proof that ...
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3answers
31 views

Linear Independence for different fields

I have a statement for a space over $R^n$: {x, y, z} is linearly ind. $\implies$ {x + y, x + z, y + z} is linearly independent Quick proof: a(x+y) + b(x+z) + c(y+z) = 0 $\implies$ (a+b)x + ...
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1answer
60 views

Why is $\left(\mathbb{Z}_{51}\right)^* \cong \mathbb{Z}_2 \times \mathbb{Z}_{16}$?

I have to show that $\left(\mathbb{Z}_{51}\right)^* \cong \mathbb{Z}_2 \times \mathbb{Z}_{16}$. I know that $\mathbb{Z}_{51}\cong\mathbb{Z}_3 \times \mathbb{Z}_{17}$ and that $(\mathbb{Z}_p)^*\cong ...
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What are the properties of a prime number?

For instance, we know that odd numbers behave like: $$x = 2y + 1 \quad\text{where}\quad x,y\in\mathbb Z$$ For even numbers: $$a = 2b \quad\text{where}\quad a,b\in\mathbb Z$$ But what about prime ...
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1answer
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Prime values of binomial

Does there always exist $x$, such that $x>b$ $x>a$ and $a+bx^n$ is prime? Of course, $a$, $b$ are given relatively prime numbers. I know that is true for n=1 in general, and I understand that it ...
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Infinite and Finite Sets of Primes [closed]

The set of all primes is infinite. Here is the list of some sets of primes with additional structure: Real Eisenstein primes: $3x + 2$ Pythagorean primes: $4x + 1$ Real Gaussian primes: $4x + 3$ ...
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2answers
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What would be the consequences of proving Riemann's hypothesis for Legendre's conjecture?

I've heard somewhere that Riemann's hypothesis doesn't imply Legendre's conjecture. But if Riemann's hypothesis is true, would an interval maybe a bit larger than $[n^2,(n+1)^2]$ contain always at ...
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1answer
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Numbers used for modeling impose constraints on the model

In modeling observation we use different numbers. Mostly either positive integers or rationals. Both impose constraints on the model description. Positive integers have exactly one minimal element ...
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Show that $16$ is a perfect $8$th power modulo $p$ for any prime number $p$ [duplicate]

Show that $16$ is a $8th$ power $\mod{}$ $p$ for any prime number $p$. I have no idea how to approach this. I tried, $$a^8\equiv16\pmod{p}$$ $$(a^4+4)(a^4-4)\equiv 0 \pmod{p}$$ $$a^4 \equiv ...
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1answer
25 views

Why $x \le (1+\frac {ln(x)} {ln(2)})^{\pi(x)}$ imply $\pi(x)\ge \frac {ln(x)} {2lnln(x)}$ for $x \ge 8$?

Let $\pi(x) = |\{ p \le x : p \in P\}|$ denote the prime counting function $\pi:\mathbb R \rightarrow \mathbb N$ and $P$ the set of primes. The equality $$x \le \lfloor \prod_{p_i\le x} 1+\frac ...
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0answers
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Lower bound for twin prime counting function

We know an upper bound for twin prime counting function. If i'm not mistaken it was (x*loglogx)/(logx*logx). Do we have a lower bound for it? Is it the same function? I mean is there c constant such ...
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Find a formula for an almost-prime based sequence

Let Z(n,k) be the $n$-th $k$-almost prime. Find R(n) which is equal to Z(n,k) when k is minimal and for all l>0, Z(n,k+l) > 2^l * Z(n,k). First elements: 2, 3, 9, 10, 27, 28, 30, 81, 84,..
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1answer
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About Brun's Theorem

http://arxiv.org/pdf/1401.7555.pdf On the page 8 there is a proof of Brun's theorem. $$\large-\int_1^{\infty}\pi_2(x)\;\mathrm{d}\left(\frac1{\lfloor x \rfloor}\right)=-\sum_{n\ge ...
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0answers
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Prime number greater than n

Consider the follwing problem: Given $n$ (in binary) output a prime number $p \geq n$ (not necessarily the first prime number after $n$) Are there better techniques than the trivial one that scans ...
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3answers
42 views

Does this function describing super-primes converge?

The prime numbers are positive integers that have no multiplicative structure. One method for counting the number of primes contained in a positive integer is sieving. As an example, for the integer ...