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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Primes of the form $x^2+n\cdot y^2$, given $n$?

In an attempt to get to grips with algebra for a course I intend to follow, I was working through a bunch of exercise sheets. A series of questions got me wondering: Given an integer $n$, is there ...
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$\tau$ and grouping of prime numbers

From Prime Number Theorem and this we can state $$\frac{p_n}{\bar{p}}\sim 2$$ or $$\lim_{n\to \infty} \frac{np_n}{(p_1 + \dots +p_n)} = 2$$ If we then look at the fluctuations in the graph of $$f(n) ...
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42 views

“Race” of the primes modulo $1,3,7,9\ \pmod {10}$

The "race" starts with the prime $11$. The number of primes $1, 3, 7, 9 \pmod {10}$ is denoted after every occurring prime. Does the lead change infinitely often? And does every "runner" have ...
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1answer
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How many ways can a quadratic form represent a prime?

Given $a,b,c,p\in\Bbb N$ with $b^2-4ac<0$ and $p$ is a prime with $\bigg(\frac{b^2-4ac}p\bigg)=1$, how many solutions $(x,y)\in\Bbb Z^2$ are there to $$ax^2+bxy+cy^2=p?$$
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Conjectured new primality test for Mersenne numbers

How to prove that this conjecture about a new primality test for Mersenne numbers is true ? Definition: Let $M_{q}=2^{q}-1 , S_{0} = 3^{2} + 1/3^{2} , \ and: \ S_{i+1} = S_{i}^{2}-2 \pmod{M_{q}}$ ...
3
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1answer
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+50

Conjectured primality test for specific class of $N=k\cdot 6^n-1$

How to prove that this conjecture is true ? Definition : $\text{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)~ , \text{where}~ m ~\text{and}~ x ...
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2answers
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Does every prime of the form $4k+1$ divide a number of the form $4^n+1$?

While playing around with Fermat's little theorem I was asking myself the question in the title and I can't answer it...
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2answers
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2
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1answer
50 views

Prove that $\sum_{t=1}^{p-1} \frac{t^2-1}{t^2+1} \equiv 0 \pmod p$

I'm trying to prove the statement $\sum_{t=1}^{p-1} \frac{t^2-1}{t^2+1} \equiv 0 \mod p$ and I don't really know where to start. Obviously $\sum_{t=1}^{p-1} \frac{t^2-1}{t^2+1} = 2\sum_{t=1}^{(p-1)/2} ...
3
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1answer
74 views

Which constellations of primes recur forever?

Having derived much joy and learning from the answers I have received to four previous questions, let me ask one more. Let a constellation of primes be a set of primes that stand in certain fixed ...
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3answers
135 views

What's the smallest known interval containing at least one prime number?

Wikipedia says that Dusart proved in 2010 that there's at least one prime between $x$ and $\left(1 + \frac{1}{25\ln^2x}\right)x$ for $x \geq 396738$. For $x_0 = 396738$, this implies a prime between ...
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2answers
45 views

Suppose $p, p+2, p+4$ are prime numbers. Prove that $p = 3$ not using division algorithm.

Suppose $p, p+2, p+4$ are prime numbers. Prove that $p = 3$ not using division algorithm. Hint: why can't $p = 5$ or 7? So I have done the two hints and in both cases I get a 9 in my set of numbers, ...
0
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1answer
31 views

Lower bound for $\Pi(n)$ - viability of probabilistic theory

Can somebody check the validity of my arguments below, and tell me why its wrong or right? Consider the sequence of non-negative integers. Let $a_0=0, a_1=1, ..., a_i=i,...$ Divisiblilty of $a_i$ ...
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1answer
28 views

Can Stirling's approximation be used to obtain lower and upper bound for $\pi(x)$?

The Willan's formula is given as follows (taken from Ribenboim's Little book of bigger primes): $$ \pi(x)=\sum_{j=2}^{x}f(j) \text{ where } ...
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1answer
54 views

Proving that and how $ \frac{1}{n}\sum\limits_{p\le n}\lfloor n/p \rfloor - \sum\limits_{p\le n} 1/p $ approaches $0$

Let $p$ denote a generic prime number. By Mertens' second theorem, the sequence $$\sum\limits_{\ p \le n} \frac1p - \log\log n$$ converges to the Meissel-Mertens constant $M\approx 0.2614972$. Now let ...
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33 views

Trial division formula [on hold]

I'm not very good with mathematic notation, I'm trying to describe a formula for trial division. Can anyone point me in the right direction or provide an answer? Also please do explain the notation ...
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0answers
15 views

Primes and even numbers [duplicate]

Can every even number be written up as the difference of two primes? And in either case could you prove it? $$ e.g.\space 8 = 31-23 \space and \space 10 = 41-31$$
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28 views

Ways Of Finding Primes and If they are efficient

I am currently in middle school and love number theory. I try and do a proof every day and today I was working on a relatively simple one involving primes. I proved that every prime above 5 can be ...
2
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1answer
73 views

Why is $p_n \sim n\ln(n)$?

I know that the prime number theorem states that the number of primes less than $x$ is approximately $\frac{x}{\ln(x)}$. However, why does this mean that $p_n \sim n\ln(n)$? (where $p_n$ is the $n$-th ...
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2answers
305 views

The longest sequence of numbers with a certain divisibility property

Definition - Denizen A sequence $\lbrace a_k \rbrace$ is a denizen if all of it's members are prime numbers, i.e $a_0, a_1, ... a_n \in \mathbb{P} $; and it satisfies the following condition; ...
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2answers
98 views

How many prime numbers we need? [on hold]

If we have some not prime number $n > 1$ we always can make prime factorization. For this operation we need $m$ prime numbers. Is there any way to prove that for given $n$ we can use no more then ...
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Algorithm to find primes up to $n$ in $O\left(\frac{n}{\log n}\right)$?

Consider the problem of given an integer $n$, generating a list of the primes not greater than $n$. An optimized version of the Sieve of Eratosthenes can do such task in $O(n)$, while the more modern ...
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The distribution of prime and semi-prime.

Let $\alpha$ be an integer and $\rho_1,\rho_2$ some prime such that $\alpha=\rho_1\cdot\rho_2+1$, and $\beta$ the number of all semi-prime less than or equal to $\alpha$. Prove ...
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2answers
86 views

New deterministic primality test for numbers of the form $p\cdot 2^n + 1$

Edit: Sorry, there was an error. Old Claim (not true because there is a counter-example): Let $p$ be prime. Let $n \in \left\{1, 2, 3, ...\right\}$. Then $N = p\cdot 2^n+1$ is prime, if and only ...
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1answer
46 views

Show that there exists $s, t \in S$ such that $\gcd(s, t)$ is a prime

Let $S$ be a set containing finitely many positive integers greater than 1 with property: for all $n \in \mathbb{Z_+}$, there exist $s \in S$ such that $\gcd(s, n) = 1$ or $\gcd(s,n) = s$. Show that ...
2
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2answers
50 views

Is the sequence $\{0,2,6,12,20,30,…,n(n+1)\}$ admissible for every natural $n$?

Look here : https://en.wikipedia.org/wiki/Prime_k-tuple for the definition of an admissible sequence. I wonder if the sequence of differences of primes can be $\{0,2,4,6,8,...,2n\}$ for every ...
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1answer
25 views

Dirichlet theorem

Can anyone give a simple number theory proof for the Dirichlet theorem? Statement of Dirichlet theorem:given any two numbers a and b whose g.c.d is 1,Prove that infinitely many primes exist in the ...
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2answers
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How do I find(isolate) the n-th prime number?

So I wanted to solve this SPOJ problem and I did some research about finding the n-th prime number. This formula came across and it stated that the n-th prime must be in this range: $n \ln n + ...
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what is formula to this eqution [(256)16]1/32+[(169)6]1/12 [on hold]

how to solve this equation [(256)16]1/32+[(169)6]1/12 what is formula of this? What is the closed form expression for this? What is the right domain for this Hamiltonian 2? what is the right ...
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4answers
72 views

Books on Prime numbers

I am a graduate student and have just finished Burton's book on number theory. Now I want to read further on prime numbers. Does anyone have any suggestion?
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A natural number has exactly 10 divisors including 1 and itself.How many distinct prime factors can this natural number have?

A natural number has exactly 10 divisors including 1 and itself.How many distinct prime factors can this natural number have? options given: (a) either $1$ or $2$ (b)$1$ or $3$ (c)either $2$ or ...
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1answer
55 views

Is every sufficiently large even integer the sum of distinct primes?

Is every sufficiently large even integer the sum of (any number of) distinct primes? No doubt this question has been asked before; does the conjecture/theorem have a name? It is related to ...
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1answer
55 views

Smallest twin-prime-pair exceeding $10^{1000}$

I found the twin-prime-pair $$\large 10^{1000}+9705092\pm 1$$ with PARI/GP. Is this the smallest twin-prime above $10^{1000}$ ? A general question to the search of twin primes : The prime number ...
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Is there a tighter approximation for the least prime gap of a given length?

This link https://primes.utm.edu/notes/gaps.html gives a definition of the maximal gaps. For a number $g$ , $p(g)$ is the smallest prime $p$ followed by at least $g$ composites. The estimate is ...
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23 views

Chebyshev's original proof of Bertrand's postulate

I'm looking for the original Chebyshev's proof of Bertrand's postulate. It would be great if someone could provide me the link to the article. Thank you in advance,
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1answer
29 views

Is there a number which describes the approaching ratio of twin primes to other primes? Or a formula for the change in density of twin primes?

Could someone shed some light on what we know about the density of twin primes? I find that it seems to be empirically true that the density of prime gaps increases as $\log(x)$ does for any gap. ...
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1answer
35 views

For every natural integer $N>3$ there are at least two distinct prime numbers $p$ and $q$ such that $\dfrac{p+q}{2}=N$ and $N-p=q-N$, $(p<q)$.

I'm not sure but this problem may be similar or related to Goldbach conjecture? Any proof/disproof, insight and opinion is appreciated, thanks.
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3answers
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If $p$ is a prime and $p \mid ab$, then $p \mid a$ or $p \mid b$.

The proof is already given in the textbook but I tried other way around. Proof by contradiction: Let's assume that $p$ doesn't divide $a$ and $p$ doesn't divide $b$, but $p$ divides $ab$. So ...
6
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8answers
353 views

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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2answers
439 views

For every positive integer n, the number $a^{2^n}−1$ has at least n+1 distinct prime divisors

Let $a>3$ be an odd integer. Prove that for every positive integer $n$, the number $a^{(2^{n})}-1$ has at least $n+1$ distinct prime divisors. This problem smells very strongly of induction, but ...
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1answer
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Large pairwise coprime sets

Say that a set $S\subseteq\Bbb N$ is pairwise coprime if every two elements of $S$ are relatively prime. Denote by $f(n)$ the size of a maximal pairwise coprime subset of $\{1,...,n\}$. What is ...
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1answer
269 views

Unique Decomposition of Primes in Sums Of Higher Powers than $2$

Primes of the form $p=4k+1\;$ have a unique decomposition as sum of squares $p=a^2+b^2$ with $0<a<b\;$, due to Thue's Lemma. What is known about sums of $n$ higher powers resulting in ...
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2answers
196 views

What is the smallest prime $p$ such that the next prime is greater than $p+2000\ $?

I studied this site https://en.wikipedia.org/wiki/Prime_gap and wondered if the smallest prime gap greater than $2000$ can still be determined, in other words : Which is the smallest prime $p$, ...
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5answers
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What is so interesting about the zeroes of the Riemann $\zeta$ function?

The Riemann $\zeta$ function plays a significant role in number theory and is defined by $$\zeta(s) = \sum_{n=0}^\infty \frac{1}{n^s} \qquad \text{ for } s > 1 \text{ and } s= \sigma + it$$ The ...
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Kronecker symbol vs. Koblitz symbol

In Koblitz, Introduction to Elliptic Curves and Modular Forms on page 188 it is defined $$\left( \frac{-1}{j}\right)=0$$ in case $j$ is even. Apart from that definition $\left( \frac{c}{d}\right)$ is ...
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Does anyone recognize this graph?

It's a plot of the following: Let $$f_{(n)} = \frac{np_n}{(p_1 + \ldots + p_n)}$$ so that $$g_{(n)} = \left|\space f_{(n)} - f_{(n-k)}\right| $$ where $n > k$ and $k = 5$ in this example. For ...
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Why does $\frac{x^n}{n^x}$ stop growing at the approximate value of $\pi (n)$?

I noticed while playing around with these functions that $n^x$ will start slow and then speed up really fast in its growth rate. While $x^n$ grows more slowly, but faster than $n^x$ at the start. ...
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2answers
74 views

Proving the irrationality of the concatenation of the $n$th powers of primes

Note: The apostrophes are meant to separate different groups of digits. Like, $0.{1^2}'{2^2}'{3^2}'{4^2}'\cdots=0.14916\cdots$. I wasn't able to come up with something better. It is easy to show ...
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1answer
62 views

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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2answers
62 views

Proof that every positive integer has at most one prime factor greater than it's square root?

I read the statement in the title somewhere but I can't find any proof. For a positive integer $n$, why can't there be 4 numbers $a, b, c, d$ ($b$ and $d$ are prime) for which $a < \sqrt{n} < ...