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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Finding all the divisors of $a$ by decomposing it into the product $p^{\alpha_1}_{1} \cdot p^{\alpha_2}_{2} \cdots p^{\alpha_r}_{r}$

I'm trying to prove the following statement regarding the fundamental facts of prime numbers, but I don't really understand the relationship between $a$ and $b$. In order to find all the divisors ...
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What is the next prime with this form?

The following are primes, $$P_1 = 2^2 + 3^3$$ $$P_2 = 2^2 + 3^3 +5^5 + 7^7$$ After these two, the only prime of such form I've found is, $$P_3 = 2^2 + 3^3 + 5^5 + 7^7 + 11^{11} + 13^{13} +\dots+ ...
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391 views

Prime Numbers and a Two-Player Game

In this question, $\mathbb{N}_0$ is the set of all nonnegative integers. The notation $\mathbb{N}$ is reserved for the set of all positive integers. Alex and Beth are playing the following game. ...
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230 views

Inequality involving a product over the primes

Is there someone who is able to prove the following statement? $$\prod_{m=1}^n \dfrac{p_m-1}{p_m} \leq \dfrac{1}{\ln(n)}$$ for all integers $n >1$ where $p_m$ is the $m$-th prime number.
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Is $a=\frac{1992!-1}{3449\times 8627}$ a prime number?

Is $a=\dfrac{1992!-1}{3449\times 8627}$ a prime number ? This is a natural follow-up to that recent MSE question We know that $a$ has $5702$ digits and no prime divisor $<10^6$.
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1answer
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inequality on Gaussian prime

Consider any Gaussian prime $p$ (except $|p|=\sqrt{2}$). If we have $|x|\leq|p|+0.5$, where $x$ is a nonzero Gaussian integer, can we prove $|x|\leq|p|$?
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Summation of prime multiples less than n [on hold]

How can I sum the following $$ \sum (2^x\cdot3^y\cdot5^z\cdot7^w\cdot\prod_1^m(p_i^{a_i})) $$ with $$2^x\cdot3^y\cdot5^z\cdot7^w\cdot\prod_1^m(p_i^{a_i}) \le n$$ where $p_i \ge 11$ are list of fixed ...
2
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1answer
55 views

How to show $n$ is a prime number?

Let $a$ and $n$ be integers greater than 1. Suppose that $a^n-1$ is a prime. Show that $a=2$ and $n$ is a prime. What can you say about primes of the form $2^n+1$? By ...
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0answers
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Are logarithms of prime numbers algebraically independent?

From Baker's theorem it follows that a linear combination of natural logarithms of prime numbers with non-zero algebraic coefficients can never be zero. Has it been proved that the set of all natural ...
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1answer
431 views

Is $1992! - 1$ prime?

Consider the factorials, defined inductively by $1! = 0! = 1$ and $n! = n\cdot(n-1)!$ for $n \geq 2$. Question: Is $1992!-1$ a prime number? The question is from a book, maybe is contest math ...
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Is the Green-Tao theorem valid for arithmetic progressions of numbers whose Möbius value $\mu(n)=-1$?

I am reading the basic concepts of the Green-Tao theorem (and also reading the previous questions at MSE about the corollaries of the theorem). According to the Wikipedia, the theorem can be stated ...
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0answers
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+50

A lower bound for an arithmetic function

Let $N \in \mathbb{N}$ such that $\phi(N) \sim N$, where $\phi$ is the Euler's totient function. Let $A \subset [N] := \{1, 2, \ldots, N\}$. For $n \in \mathbb{N}$ define the function $$ C_A(n) = \#\{ ...
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3answers
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Upper bound for prime-counting function: $ \pi(n)\le\frac{n}{3}+2 $

$ \pi(n)\le\frac{n}{3}+2 $... Could someone explain me, how to prove it? I'm completely stuck, as informations I found on Wikipedia aren't very clear to me. (I was able to prove that for sufficiently ...
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0answers
15 views

Prove that if $\ p^2 = a^2+2b^2 $ then $\ p = m^2+2n^2 $ (where a, b, m, n are integers, and p is prime) [duplicate]

Given that $\ p^2$ can be written in the form $\ p^2=a^2+2b^2 $ (where a & b are integers, and 'p' is a prime number), then prove that the prime number 'p' can also be written in the form $\ ...
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2,1,3,5,7=???????? i need help plz [on hold]

I cannot figure this out! I know nothing about prime numbers in sequence anymore. Ive been out of school for a long while now.
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2answers
355 views

Proving that if $p^2 = a^2 + 2b^2$ then also $p$ can be written in form $a^2 +2b^2$

I'm high school student, and this problem has bothered me for about 2 weeks now. I don't necessarily need a solution, but for example mentioning a helpful theorem or property that could help me to ...
6
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3answers
70 views

sequence of primes in arithmetic progression

The question is: Suppose $p_1<p_2<...<p_{15}$ is a sequence of prime numbers in arithmetic progression, with common difference $d$. Prove that $d$ is divisible by $2,3,5,7,11$ and $13$. Let ...
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2answers
36 views

Property of additive group [closed]

Let $m \in \mathbb{Z}_q$, for a prime $q$ and $x \in \mathbb{A}$, where $\mathbb{A}$ is an additive group of order $q$. Then is it always true that $mx \in \mathbb{A}$? If true, how to prove it? ...
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1answer
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Wieferich prime-Lang-Trotter conjecture connection?

Crandall-Dilcher-Pomerance prediction states that the number of Wieferich primes $<x$ is $log\ logx $ N.Katz in "WIEFERICH PAST AND FUTURE" states; The Crandall-Dilcher-Pomerance prediction is ...
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2answers
51 views

How many number id divisible by $p$ and is not divisible by any primes number which is less than $p$?

Let $N$ be a big integer. Let $p$ be a prime number. Is there a formula to count how many number less than $N$ such that they are divisible by $p$ and not divisible by any prime less than $p$. For ...
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1answer
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In the definition of Carmichael number, why is it necessary to have $(b, n) = 1$?

In number theory, a Carmichael number is a composite number $n$ which satisfies the modular arithmetic congruence relation $$b^{n-1}\equiv 1\pmod{n}$$ for all integers $1<b<n$ which are ...
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1answer
55 views

Can a Carmichael number be even?

Can a Carmichael number be even? I know that a Carmichael number is a positive composite integer $n$ such that $a^n\equiv a \pmod n$ for all integer $a$. So what does I need to prove or disprove ...
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1answer
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ABC conjecture consequence

At page 6 of the book: "Prime Numbers The most mysterious figures in Math" this statement is listed as one of the consequences of the ABC conjecture: There are Infinitely many Wieferich primes. This ...
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3answers
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The square of n+1-th prime is less than the product of the first n primes.

I wanted to prove the following question in an elementary way not using Bertrand postulate or analytic estimates like $x/\log x$. The question is $$ p_{n+1}^2<p_1p_2\cdots p_n,\qquad(n\geq4) $$ I ...
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4answers
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Find the smallest positive integer that ends in $17$, is divisible by $17$, and the sum of its digits is equal to $17$.

This is a very interesting word problem that I came across in an old textbook of mine. So I know its got something to do with primes and composities but other than that, the textbook gave no hints ...
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3answers
71 views

Prime number equation

The number of solutions of the equation $xy(x+y)=2010$ where $x$ and $y$ denote positive prime numbers, is ____ I tried various things but nothing seems to work out. $2010$ can be resolved into ...
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473 views

Primes and certain unit fractions [closed]

Are there primes $p,q$ and a natural number $a$ such that $\frac{1}{p}+\frac{1}{q}=\frac{1}{a}$?
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Largest Arithmetic Sum of Relatively Prime Numbers Under 30

Pick however many integers in the range $[1,30]$ (inclusive). The only constraint is that all of these numbers must be relatively prime to each other. What is the largest possible arithmetic sum ...
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1answer
34 views

What does “421 is the smallest prime formed by the powers of two in logical order from right to left” mean and if so is it correct?

I've seen this on number gossip and a few other places, but I'm not exactly sure what it means. The only possibilities I have thought of for what they mean are "421 is the smallest center squared ...
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The asymptotic behaviour of triples $n!+q^{n!}=c$, where $q$ is the first prime greater than $n$, and abc conjecture

For a large positive integer $n$, let $q=q(n)$ (below we denote this $q=q(n)$ by $q_{N}$ because we assumed that is the $N$-th prime number) the first prime number which is (strictly) greater that ...
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1answer
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What does the sum of the reciprocals of composites run along?

This is fairly straight forward: $$\sum_{p\space\text{prime}}^x \frac{1}{p_x} \sim \ln(\ln(x))$$ And if $$\sum_{c\space \text{composite}}^x \frac{1}{c_x}\sim f(x)$$ Then what is $f(x)$?
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Modulus and Fermat's Little Theorem

How do I calculate $ 11^{23} \bmod{163} $ using fermat's little theorem ?
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1answer
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Counting how many primes exists between square root of a given range?

I am given with a range say l,r(1<=l,r<=10pow14).I am also given with cumulative count of primes no.'s that exists between 1 to 10pow7,e.g for 1,count=0(as no primes exists upto 1),for ...
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1answer
46 views

What is the smallest “prime” semiprime?

(All dots here means concatenation.) Let $s= ab$ be a semiprime number, then I call s a "prime" semiprime if all these following conditions are satisfied: The reversal of $s$ is a prime The ...
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0answers
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Beal Conjecture and ($\bmod 3$) operation [closed]

When we apply a ($\bmod 3$) operation on the $A^x +B^y =C^z$ we will see some strange results. For e.g.: Let $A=6m+1$ & $B=6n+1$. Since $A$ & $B$ are odd numbers, $C$ will have to be even. ...
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1answer
152 views

Number we know all prime numbers less than [duplicate]

We already know some very big prime numbers. ($2^{257,885,161} − 1$ as of time of writing is the largest known) It is my understanding, that we know it is a prime number but we don't know all prime ...
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2answers
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The number $2^{2^n} + 2^{2^{n - 1}} + 1$ can be expressed as the product of at least $n$ prime factors

Prove that the number $2^{2^n} + 2^{2^{n - 1}} + 1$ can be expressed as the product of at least $n$ prime factors, not necessarily distinct. Doing what the hint has suggested, I have done the ...
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1answer
288 views

Fibonacci $\equiv -1 \mod p^2$

Is there a prime $p > 3$ such that the Fibonacci number $F_{np} \equiv -1 \mod p^2$ for some natural number $n$? I know none of the first $1000$ primes $> 3$ qualify. EDIT: In response to ...
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2answers
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Is $\frac{2^{n-1}-1}{n}$ an integer only when $n$ is an odd prime?

I have the equation $$k = \frac{2^{n-1}-1}{n}$$ and wonder if $k$ is an integer when $n$ is an odd prime. The numerator is always odd, so even $n$ have no integer solutions. But when I run a test, it ...
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Proof of the reciprocal of all semiprimes diverging?

$$\sum_{\text{semi-primes}}\frac{1}{s}=\frac{1}{4}+\frac{1}{6}+\frac{1}{9}+\frac{1}{10}\cdots$$ I almost positive that this sum diverges, but I would really like to see a very thorough proof. Thank ...
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1answer
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find least multiple formed only of 1's of given number

The problem states that given a number find the least multiple formed only of 1's. If no such number exists then 0 will be the answer. For example for: ...
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3answers
111 views

How can $p^{q+1}+q^{p+1}$ be a perfect square?

How can one find all primes $(p,q)$ such that $p^{q+1}+q^{p+1}$ is a perfect square I considered it $\mod 2$ and found a trival solution . Im curious about an eventual answer Diophantine equations ...
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1answer
56 views

The Sieve of Alice- Number theory Riddle

I am trying to prove the result for a problem which I am unable to do so! The answer is simply $\frac{N}{2}$ when N is even and $\frac{N}{2}+1$ when $N$ is odd. But I do not see why?? Can you give me ...
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3answers
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primes whose difference is a multiple of $n$

Hello everyone this is my first question here so if there are any suggested edits feel free to participate! Is it true that for every $n$ there two prime numbers $p$ and $q$ with $n\mid p-q$? I have ...
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Is this proof for number of divisors correct?

I'm trying to prove that the number of divisors of any given number is $(a_1+1)(a_2+1)...(a_r+1)$ Where $a_1, a_2 ... a_r$ are from $p_1^{a_1}p_2^{a_2}\cdots p_r^{a_r}$ The problem is that the proof ...
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2answers
48 views

Summation of a series of Positive Prime numbers

Is there a way to find the sum for a set of positive prime numbers (e.g., the first $25$ prime numbers) without just arbitrarily adding them up shorthand?
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Is (73, 37) the only pair of reversible primes (p, q), s.t. p=2q-1?

In addition to being probably the only Sheldon Cooper prime, $73$ is a reversible prime $p$ (or emirp), such that its reverse is $q=(p+1)/2$. It is not hard to see that all other reversible primes ...
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1answer
32 views

Understanding Wright's proof of Landau's theorem

I'm reading Wright's A simple proof of a theorem of Landau in which the core argument is a proof by induction and I find myself stuck on a major point. I must be misunderstanding notation or something ...
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2answers
82 views

Do the sum of all prime reciprocals with the digit $3$ converge or diverge?

$$\frac{1}{3}+\frac{1}{13}+\frac{1}{23}+\frac{1}{31}+\frac{1}{37}+\frac{1}{43}\cdots$$ Intuitively, I feel that this sum converges, but I really don't know why, (or if I am correct). Can I have a ...
0
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1answer
57 views

Out of all the proofs of the PNT, which one is the most accessible?

I have been studying the continuation of the Riemann zeta function $\zeta(s)$ for the past while. I can prove that all the zeroes must lie in the critical strip.I am currently in the process of using ...