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.

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proving theorem about perfect powers

Im currently studying the journal entitled Perfect Powers with All Equal Digits but One theorem: For a fixed integer $l \geq 3$, there are only finitely many perfect $l$-th powers all whose digits ...
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Enumeration of primes

Given a prime number $p$, there is an associated number $n(p)$, giving its ranking in the sense that $n(2)=1$, $n(3)=2$, $n(5)=3$ etc. Is there a closed form expression for $n(p)$ in terms of $p$?
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Conjecture about primes and the factorial

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 ...
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3answers
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The arithmetic function $\lambda(n)=(-1)^{a_1+\cdots +a_k}$

Define $\lambda(1)=1$, and if $n=p_1^{a_1}\cdots p_k^{a_k}$, define $$\lambda(n)=(-1)^{a_1+\cdots +a_k}$$ How can I see that $$\sum_{d\mid n}\lambda(d)=\begin{cases} 1 \,\,\text{ if $n$ is a square}\\...
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More claims deduced from even perfect numbers, and what is the meaning of such calculations

I show the following examples of easy deductions for even perfect numbers $n=2^{p-1}\cdot(2^p-1)$, where $b=2^p-1$ is its associated Mersenne prime, and after I am asking to you about the $\Leftarrow$ ...
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1answer
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Conjecture concerning modular arithmetic

Below $0\notin\mathbb N$. I want a proof or a counter-example of the following (corrected) conjecture: Suppose $p$ is the smallest prime dividing $n\in\mathbb N$ and suppose $kn+ap=m!$, where $...
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The Porsche prime

A friend told me that the number starting with 911 followed by 911 zeros ending with 119 (that is $911\cdot 10^{914}+119$) is a prime number, the so-called Porsche prime. Maple indeed confirms that ...
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1answer
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Existence of a Repeating Divisor

I have $n$ integers $a_1, a_2, a_3, .., a_n$ let $X = a_1*a_2*a_3*...*a_n$. I want to know a single integer $F$ such that $F^2$ divides $X$. It is told that there will be atleast one such $X$ and ...
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1answer
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Why is it not known if Mill's constant is rational or irrational?

The following text appears in the Mill's constant definition at the Wikipedia: There is no closed-form formula known for Mills' constant, and it is not even known whether this number is rational (...
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The eventual advantage of a primality test without known exceptions

The primality test of Fermat with base $2$ seems to be as secure as the computer hardware for testing numbers big enough. However, I think there are an infinite numbers of false primes using this ...
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Is the Fermat primality test secure enough for very big numbers?

The random variable $X_m$ is the number of trials before $n\notin\mathbb P\wedge n|2^{n-1}-1$ where $n$ is an odd random integer $2^{m-1} < n < 2^m$. Computer simulations makes me believe ...
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Proving a number is Carmichael

here is my question: Let $p>3$ be prime, s.t $q = 2p-1$ and $g = 3p-2$ are primes as well. (For example $p=19$,$13$,$7$). Prove that $N = pqg$ satisfies $p-1|N-1$, $q-1|N-1$ and $g-1|N-1$. I ...
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Describe the prime elements of the ring $\mathbb Z[\sqrt{-2}]$ [duplicate]

I have a ring $\mathbb Z[\sqrt{-2}]$ and I need to describe all the prime numbers of that ring. How I can do that? Thank you
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1answer
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A primality test using the gcd

Let $f:\mathbb{N} \rightarrow \mathbb{N}$ be defined by $$f(n) = gcd(n,\lfloor \sqrt{n}\rfloor ! \mod n).$$ Show that a) If $p$ is a prime divisor of $n$ with $p \leq \sqrt{n}$, then $p \mid f(n)...
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The asymptotic behaviour of $\sum_{1\leq k\leq N-1}\int_{p_k}^{p_{k+1}}\log x d[x]$, where $p_n$ is the nth prime number

Let $p_k$ is the kth prime number and consider for $N\geq 2$ the arithmetic function $$f(N)=\sum_{k=1}^{N-1}\int_{p_k}^{p_{k+1}}\log(x) d[x]$$ where $[x]$ is the integer part function (provide us in ...
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1answer
33 views

Question about distribution of primes

The following is from "Introduction to Number Theory" by Hardy and Wright. The book first states the following theorem Theorem A: If $\pi(x)$ is number of primes not exceeding $x$ then $\pi(x) \...
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1answer
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What are the different ways to get a first-order formula that express the statement“$P$ is the $n$-th prime”

I know that such a $2$-predicate formula exists since Enderton's have already constructed such a formula in his text on mathematical logic but it was not easy to remember so I wonder if there is other ...
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2answers
542 views

What is the first 21-digit prime number after the decimal point of Pi

We know the recent computers are 64-bit, and the maximum integer number is 18446744073709551615, whether you can find the first 21-digit prime number after the decimal point of $\pi$? Please show me.
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1answer
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What is the largest prime $p$ such that the decimal expansion of $1/p$ repeats with period 2017?

By this discussion on John Baez's Google+ feed, the primes $p$ such that the decimal expansion of $1/p$ repeats with period 2017 are exactly those primes which occur in the prime decomposition of $10^...
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Prime Powers and Differences of Consecutive Cubes

I am wondering if it has been proven that there does not exist a prime $p$ and an integer $r \ge 3$ such that $p^r = (n + 1)^3 - n^3$ for some integer $n$. Note that this is a special case of Beal's ...
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1answer
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Is there a prime of the form $11^k+k^{11}\ $?

Is there a natural number $k\ge 1$, such that $11^k+k^{11}$ is prime ? I checked the numbers upto $k=3000$ and did not find a prime number. On the other hand, for $k=76$ and for $k=142$, there is no ...
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Generalizing the growth of sums of two squares

Consider the set $S$ of numbers which are the sum of two (integer) squares, and define $S(n)$ as the number of members of $S$ in $\{1,2,\ldots,n\}.$ It is well-known that $$ S(n) \sim \frac{Kn}{\sqrt{\...
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Notation for “the highest power of $p$ that divides $n$”

If $p$ is a prime and $n$ an integer, is there a standard or commonly used notation for "the highest power of $p$ that divides $n$"? It's a concept that is often used repeatedly in number-theoretic ...
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Is there an identity related to $\binom{n-j-1}{k}+\binom{k+j}{k}\pmod{n}$?

I noticed that when $n$ is an odd prime, the following congruence $$\binom{n-j-1}{k}+\binom{k+j}{k} \equiv 0 \pmod{n}$$ holds for $0 \le j \le \frac{(n-k)}2$ and odd values of $k$ such that $0 < k ...
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2answers
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Computing a double integral with applications to prime numbers

I was reading the preprint [1] which contains on p. 7 the following formula (for $4<s\le6$): $$ f_1(s)=\frac{2e^\gamma}{s}\left\{\log(s-1)+\int_4^s\int_3^t\frac{\log(u-2)}{u-1}du\,dt \right\} $$ ...
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8answers
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If a prime $p\mid ab$, then $p\mid a$ or $p\mid b$

If a prime number $p$ is a divisor of a product $ab$, $p$ has to be a divisor of $b$ or $a$. How can I demonstrate this theorem? I demonstrated this theorem on one way using Bezout's theorem in an ...
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There are at least two solutions such that $2p_n=p_a+p_b$ ($p$ being prime)

I've stumbled across this playing around and summing primes at random during a boring lecture. Is this a known conjecture? Can it be proven? My conjecture: There exists at least one non trivial ...
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Uncertainty in a theorem about Sarrus numbers

From https://oeis.org/A001567 there is a theorem of Ray Chandler formulated: An odd composite number $2n + 1$ is in the sequence if and only if multiplicative order of $2\;(\text{mod}(2n+1))$ ...
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Function check-up exercise

I want to make sure I did everything correctly, so here's the exercise: Given $P$ the set of positive prime numbers and be $S = \mathbb N^* - \{1\}$. $\forall n \in S,\ \pi(n)$ is the set of the ...
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Is there a reason for the existence of prime numbers? [closed]

Is there a reason for the existence of prime numbers, or is there a reason that some numbers are prime numbers, but others are not? Does the number theory have any answers or at least ideas about ...
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Find all N in $\phi(N)=98$ [closed]

Solve the equation $\phi(N)=98$ I have no idea how to do it. How to find all N?
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2answers
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Legendre symbol $(-21/p)$

I am a bit confused with the question: For what prime $p$, $\left(\frac{-21}{p}\right) = 1$? I did something like that: $$\left(\frac{-21}{p}\right) = \left(\frac{-1}{p}\right)\left(\frac{3}{p}...
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Prove or disprove that ${F_{n}^2} + 43$ is always a composite

This is a kind of follow-up to another question, but in order not to burden that question and its answers with new comments, I decided to create this separate question. Also, it looks this problem is ...
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1answer
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Function exercise check-up

I want to make sure I did everything correctly, so here's the exercise: Given $P$ the set of positive prime numbers and be $S = \mathbb N^* - \{1\}$. $\forall n \in S,\ \pi(n)$ is the set of the ...
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1answer
80 views

Are there infinitely many pairs of primes, $p$ and $q$, such that $q = 4p + 1$?

How close can one come to proving that there are infinitely many primes, $p$ and $q$, such that $q = 4p + 1$? The idea for this question came from reading the question and answers posed by user39898,...
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1answer
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Primes is in P, proof of hendrik Lenstra Jr. lemma

In the paper describing AKS primality test : http://annals.math.princeton.edu/wp-content/uploads/annals-v160-n2-p12.pdf On page no. 8 Lemma 4.7 last paragraph, I cannot understand how number of ...
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1answer
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Is the sum of coefficients 2?

Is the sum of the coefficients of the polynomial interpolation of the data $(1,p_1),(2,p_2),...,(n,p_n)$ for some positive integer $n$ (where $p_n$ is the $n$th prime) always equal to two? I've ...
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A question about arithmetic progressions and prime numbers

I took number $3$ and observed: $3$ is an arithmetic progression of length one. $3,5$ is an arithmetic progression of length two. $3,5,7$ is an arithmetic progression of length three. Then I took ...
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3answers
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A Formula For Primes

Could someone explain me why this arithmetic of sets can not be called a Prime Numbers formula? Was it already found before and is not relevant? Prime numbers sequence $\mathbb P$ (or set of members ...
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How to check if a number is prime? [closed]

I am having a problem with those numbers: 1) $2015^7 - 1$ 2) $817^2 + 53^2$. Especially when number is raised to a given power. My solution for the second point: $817^2$ is the same as checking $...
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What is meant by strictly in this statement?

If $n$ is prime, then $n$ is not divisible by any prime number between 1 and $\sqrt{n}$ strictly. (Assume that $n$ is a fixed integer that is greater than 1.). I searched online and found that "...
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If $b^2 \equiv 1 \pmod 3$, is it possible to have $\sigma(b^2) \equiv b^2 \pmod 3$?

The title says it all. Let $\sigma(N)$ denote the sum of the divisors of the positive integer $N$. To paraphrase my question: If $3 \mid \left(b^2 - 1\right)$, is it possible to have $3 \mid \...
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If $q(X)$ is reducible in $\mathbb Z[X]$, then it's reducible in $\mathbb Z_p[X]$ for every prime $p$

My book states, without a proof, that If $q(X)$ is reducible in $\mathbb Z[X]$, then it's reducible in $\mathbb Z_p[X]$ for every prime $p$. The contrapositive of the above result is more useful:...
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How to calculate $9^{47^{51}} \mod 67$?

I've looked at some other related things on here, but this seems a little more complicated with the double exponentiation. Is there a general algorithm to calculate $a^{c_1^{c_2^{...^{c_n}}}} \mod p$ ...
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1answer
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Is there anything known in general about upper and lower bounds for $\prod_{i\leq n\vee p_n>k}(p_i-k)$

I have no specific reason to ask this question other than seeing that it comes up quite often when I'm playing around with prime numbers. Let $$f(n,k)=\prod_{i \leq n\vee p_n>k}(p_i-k)$$ Where $p_i$...
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1answer
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Why can't negative numbers be prime? [duplicate]

I was in a lecture on primes, when it occurred to me what negative numbers are excluded. Why exactly?
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Prime elements of ring $\mathbb{Z}[\sqrt{-21}]$ [closed]

Find prime elements of the ring $\mathbb{Z}[\sqrt {-21}]$. Please help with some ideas.
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Sum of three consecutive prime numbers is $173$

If I tell you that the sum of three consecutive prime numbers is $173$, how quickly could we find the biggest of these numbers?
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Restricted equality involving prime numbers

Given three real numbers such that $a + b + c = 0$, it can be proved that \begin{align*} \frac{a^{5} + b^{5} + c^{5}}{5} & = \frac{a^{3} + b^{3} + c^{3}}{3}\cdot \frac{a^{2} + b^{2} + c^{2}}{2}\\ \...