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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In what quadratic or quartic integer ring is a prime of the form $a^4 + 4^b$ guaranteed to split?

The obvious choice seemed at first to be $\mathbb{Z}[\root 4 \of 4]$. But since I know next to nothing about quartic fields, I thought to look in the quadratics. For the first few such primes in ...
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1answer
26 views

Defining Primes in Non-standard Models of Peano Arithmetic

I was recently reading a post basically discussing an "intuition" that Goldbach's Conjecture may be a statement which is undecidable (the post does not specify which axiomatic system the statement is ...
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0answers
97 views

Number-theoretic asymptotic looks false but is true?

Question Let $p_r$ be the $r'th$ prime. Is it true that, $$\sum_{r=1}^\infty s^r \ln(p_r) \sim \frac{s}{(1-s)} $$ I know this looks bizarre but kindly consider the argument below. I'm also ...
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1answer
106 views

What would the Riemann Hypothesis mean for the Prime Number Theorem?

The Prime Number Theorem states $\pi(n)\sim \dfrac{n}{\ln n}$. Would there be an equally simple expression if Riemann's Hypothesis were proved true? From Chebyshev Function, would $\pi(n)\sim ...
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1answer
50 views

Unique prime factorization [duplicate]

We all know that $$15=3 \times 5$$ And $$15 =(-3) \times(-5)$$ Since $3 \neq -3$ and $5 \neq -5$ , we have two different prime factorizations ! Is this wrong ? If this is wrong , then there are ...
5
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6answers
577 views

New largest prime number discovery - what's all the fuss [duplicate]

So I've read about the latest largest prime number discovery (M74207281), but I find it hand to understand what's the big deal because using Euclid's proof of the infinitude of primes we can generate ...
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1answer
61 views

Any even is the sume of two primes [duplicate]

How can you prove or disprove that any even number is the sum of two primes?
5
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2answers
141 views

If $a^4 + 4^b$ is prime, then $a$ is odd and $b$ is even.

We say an integer $p>1$ is prime when its only positive divisors are $1$ and $p$. Let $a$ and $b$ be natural number not both $1$. Prove that if $a^4+4^b$ is prime, then $a$ is odd and $b$ is even. ...
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3answers
36 views

Finding the prime number $n$: why checking for a divisor between 2 and $\lfloor \frac{n}{2} \rfloor$ is enough?

Let's say I want to check whether 33 (say $n$) is a prime number or not. Instead of checking whether 33 is divisible by a number between 2 and 31 or not, it is sufficient enough to verify that 33 is ...
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0answers
28 views

Prove the limit $\lim_{n\to\infty}v(n)/n = 0$ where $v(n)$ is the number of prime factors of the integer $n$

Let $v(n)$ be the number of prime factors of the integer $n$. For example, $v(8) = 3$ and $v(5) = 1$. Prove: $$\lim_{n \to \infty} \frac{v(n)}{n} = 0$$ I was thinking about the sandwich theorem, just ...
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3answers
385 views

How, if at all, does pure mathematics benefit from $2^{74207281}-1$ being prime?

So a couple of days ago the $17$ million digit number $2^{57885161}-1$ was beaten by the $22$ million digit number $2^{74207281}-1$ at being the largest known prime number. Are there any specific ...
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1answer
20 views

Algorithm for factoring large decomposable primes into Gaussian primes

Given a prime $p$ (with residual 1 modulo 4) what is the most efficient algorithm for computing its Gaussian prime factors, assuming $p$ could be large (i.e. perhaps more than 100 bits). ...
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1answer
215 views

Will sums of infinitely many primes ever fail to generate almost all natural numbers?

Example Suppose we want to show that all natural numbers may be generated by summing prime numbers, then the proof may be as follows: In case the natural number in question is even, it must be a ...
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3answers
63 views

Infinitely many primes of the form $3k+2$

Prove that there are infinitely many primes of theform $3k+2$ I tried so: Let $$A= \left\{ p \in \mathbb{P} : \; p=3k+2 \right\}$$ Suppose, that the set $A$ is finite, i.e. $$A= \left\{ p_1 , p_2, ...
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3answers
42 views

$\operatorname{gcd} \, (a,b) = 1$ then $\operatorname{gcd} \, (a^n,b^k) = 1$

Statement: Suppose $(a,b) = 1$ then $(a^n,b^k) = 1$ for $n,k \geq 1$. Attempt at Proof: Let $P$ be the set of all primes. Let $P_a$ be the set of primes $p_i$ such that $$a = \prod_{i=1}^{r_1} ...
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0answers
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What is an upper bound for number of semiprimes in the interval $[n^2,n^2+2n]$

A semi prime is a number which is product of two distinct primes. What is an upper bound for number of semi primes in the interval $[n^2,n^2+2n]$?
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0answers
54 views

What's the smallest number that we don't know if it's prime or composite? [duplicate]

What's the biggest $n$ such that for all $1<x<n$, we know for sure if $x$ is prime? The smallest primes are easy to find, and the biggest ones we haven't found yet. At the top, we have Mersenne ...
3
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0answers
95 views

Combining Firoozbakht's conjecture and abc conjecture

Firoozbakht's conjecture states that for all $n\geq 1$ $$p_n^{\frac{1}{n}}>p_{n+1}^{\frac{1}{n+1}},$$ where $p_k$ the kth prime number. By asumption of this conjecture, for a fixed $n$, there is a ...
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1answer
34 views

Consecutive prime terms in an arithmetic sequence

I'm looking for "long" arithmetic sequences that contain only prime (positive) numbers. For example: $$7,37,67,97,127,157$$ Is there any known bound for the length of these kind of sequences? Is it ...
2
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1answer
40 views

What is an upper bound for number of semiprimes less than n?

A semi prime is a number which is product of two distinct prime number. What is an upper bound for number of numbers in the form pq less than n? $p,q$ are prime numbers smaller than $n$.
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1answer
53 views

Does there exists a function $f$ such that for some prime $p_1$, $p_{n+1}=f(p_n)$ gives a sequence of primes?

It occurs to me that it would be very cool if for a prime $p_1$ we have $p_{2}=2^{p_1}-1$ prime, and then $p_3 = 2^{p_2}-1$ prime, and so on. This would be a sort of infinite sequence of Mersenne ...
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2answers
19 views

Mysterius semiprime fact in other number bases

So you have a semiprime n n = p*q where p < q A curious fact about bases is that if a number x ends with a zero in base y, then x is divisible by y. Therefor, if we where to represent n in all ...
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1answer
73 views

Is there a fixed integer $n$ for which ${\pi}^{n}$ is prime number?

I would like to know the relationship between $\pi$ and prime numbers distribution ,then I would like to ask if there is a fixed integer for which ${\pi}^{n}$ can be prime or how do i disproof that ...
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5answers
387 views

Which of the following is not a prime number?

Which of the following is not a prime number ? $a.)\ 911 \ \ \ \ \ \ \ \ \ \ b.)\ 919 \\ \color{green}{c.)\ 943} \ \ \ \ \ \ \ \ \ \ d.)\ 947$ This was asked in my exam and the time given per ...
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2answers
76 views

Are there any odd primes like this? [closed]

Are there any odd primes $p, q, r$ such that $$(p-1)(q-1)(r-1)\mid pqr-1$$
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1answer
44 views

Counting squarefree numbers which have $k$ prime factors?

How to find an asymptotic formula for this function below? $$f(n)=\sum_{pq\leq n}1$$ where $p$ and $q$ are different prime numbers. I guess we can write $$f(n)=\sum_{p\leq \sqrt{n}}\pi ...
3
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2answers
76 views

What is an upper bound for number of prime powers less than $n$?

What is an upper bound for number of prime powers less than $n$? I mean the numbers in the form $a^b$ in which $b \ge 2$ and $a$ is a prime number. I have found that $\frac {\log n} {\log 2} + \frac ...
3
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0answers
19 views

Product of the Euler phi function [duplicate]

Prove the following statement: If $n, m\in\mathbb{Z} $ and $g=$gcd$(n, m) $ then is $$\varphi(m, n) =\frac{ \varphi(m) \varphi(n) g} {\varphi(g)}. $$ Hint: Prove the statement with induction above ...
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1answer
60 views

Is there an non provable sentence from Peano Arithmetic?

I'm trying to deduce the following sentence using only Peano Axioms: "There exist infinite prime numbers" Since PA is known to be incomplete, its possible there is no such proof supporting the ...
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0answers
29 views

Quantitative aspect of Chebotarev Density Theorem

I recently learned the Chebotarev Density Theorem for global fields. As far as I have seen, all applications of CDT seem to focused around proving some set of prime ideals (or places in function ...
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2answers
55 views

an upper bound for number of primes in the interval $[n^2+n,n^2+2n]$

What is an upper bound for the number of primes in an interval of $n$ consecutive numbers? What is an upper bound for the number of primes in the interval $[n^2+n,n^2+2n]$?
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0answers
17 views

Nouvelle theoreme concernant les nombre premiers [closed]

Quelque soit M > 0, il existe une infinity de nombres premier dont la longueur depasse M, et qu'elles sont vides des nombrtes premiers veuillez voir ca: http://www.cjoint.com/c/FApjwGUjjWG Qu'en ...
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2answers
24 views

How do i show this :for every prime $p> 3$ and every integer $k\geq1$ ,${p}^{4k}=1\mod3$?

There are many formula which are a multiple of $3$ for example $n^3+2n$ ,I accross this formula " ${p}^{4k}=1\mod3$" after some computations in WA then My question here is: How do i show this if it ...
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2answers
68 views

Is there an interval that contains more squares of primes than primes?

The primes $p$ are, of course, in one-to-one correspondence with the squares of primes $p^2$. But is there any interval $a < x < b$ possible where the primes thin out so much, that it contains ...
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1answer
48 views

Gaps between pairs of twin primes.

I'm sorry if this has been answered before but I've not been able to find much info on it; so my question is: Say I have a pair of twin prime numbers $P_n$ and ($P_n+2$) , and the next largest pair ...
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0answers
41 views

Is the following statement true: $F_n\leq p\leq F_{n+2}$

Let be $n\in\{2,3,\ldots\}$ Is the following statement true: There exists a prime number so that $F_n\leq p\leq F_{n+2}$ while $F_n$ is a Fibonacci number.
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0answers
24 views

A question about a sequence of sets of prime numbers deduced from Euclid strategy

Let the sequence of sets of prime numbers defined by $$S_1=\{2\},$$ and for $n>1$ $$S_n=S_{n-1}\bigcup\{\text{p prime such that p divides } 1+\prod_{s_i\in S_{n-1}}s_i\}.$$ Examples. We have ...
2
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2answers
95 views

What numbers are relatively-prime to $256?$

Given the numbers are in the range $1$ to $256$, which ones AREN'T co-prime, would be an easier question$?$ This question may be very specific and hopefully trivial for somebody on the maths board, ...
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0answers
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Finding a lower bound on square free $d \equiv 1 \mod{8}$ with prime factors in arithmetic progression

I have several questions related to the problem of counting square-free $d \equiv 1 \mod{8}$, where $d \le X$ such that if $d = p_1 \cdots p_k$ then $p_i \equiv \pm 1 \mod{8}$ for all $1 \le i \le k$. ...
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4answers
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The $n$th prime number is $85489307341$. How to find $n$?

Say you are given the $n$th prime number $p_n$, like $p_n = 85489307341$, but not $n$. Questions: What's a quick, simple, and approximate formula for $n$? By adding more terms, can this formula be ...
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1answer
31 views

Is this true for every prime $p>2$ , if$ m$ is even integer number then $m$ can't be written as :$m=\prod _{i=1}^{r}{p_i}^{a_i}$?

I would like to show if $p_i$ an odd prime for all $i=1,\cdots,r$ and suppose that there is an integer $m$ such that 2 divides $m$ , I would like to show if $m$ can be written as follow: ...
5
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1answer
74 views

Does there exist a prime number $p$ such that $p^2 \mid 2^{p-1}-1$?

Does there exist a prime number $p$ such that $p^2 \mid 2^{p-1}-1$ ? I tried for some small number $p$ and I think that it does, but I don't know how to prove this.
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1answer
35 views

What is the largest prime number in the denominator of a fraction that creates a decimal that repeats every 4 digits?

I was studying a Target question for Math League competitions, and after a few hours of pondering, I finally came up with the following method of solving the mentioned problem: For any decimal, it is ...
2
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0answers
52 views

Proof that the spectrum of prime distribution will give zeros of Riemann Zeta function

All: Many of us have read that the spectrum of prime distribution will give zeros of Riemann Zeta function. For example, Mazur and Stein's book: (http://wstein.org/rh/rh.pdf ) have many nice pictures ...
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4answers
102 views

Congruences and prime numbers

I was first asked to show that a product of numbers of the form $4k+1$ also has this form. I got stuck on the next part: Deduce that if $n \equiv −1 \pmod 4$ and $n > 0$ then $n$ must have a prime ...
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1answer
11 views

Determining a homogeneous polynomial (with N indeterminates) from an integer

Imagine you have some computer program that requires an input of N values (say $a,b,c$), and calculates some homogeneous polynomial (with some small natural number coefficients) returning an integer ...
4
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1answer
77 views

Some heuristics about the Pisano Period, primes and Fibonacci primes. What reasons are behind them?

I started to read about the Pisano Period, $\pi(n)$, applied to the classic Fibonacci sequence and made some simple tests looking for possible properties of the sequence. I have observed the following ...
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0answers
17 views

Prime Counting: In truncation rule #2 mentioned in an AMS.org article, I'm unclear how “special leaves” work?

I'm reading through an AMS.org article on prime counting. The article covers the history of prime counting and focuses on improvements to the Meissel-Lehmer method. It is the improvement on page ...
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5answers
56 views

Why is the product of two consecutive integers $n \cdot( n+1) \forall n > 2$ guaranteed to have at least two prime factors?

I was reading this paper: http://fermatslibrary.com/s/a-new-proof-of-euclids-theorem and became confused when reading this line: Since $n$ and $n + 1$ are consecutive integers, they must be ...
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1answer
16 views

Asymptotic elements of a sequence of of gaps given the average asymptotic function.

If the average consecutive difference of a sequence of numbers is asymptotically the same as $f(n)$. Then what can be said about numbers in the sequence, asymptotically as $n \to \infty$. Let ...