Questions tagged [prime-numbers]

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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Renyi entropy of prime gaps

Denote with $p_n$ the $n$-th prime number and let $$ h_N(d) = |\{ n : p_{n+1} < N, p_{n+1} - p_n = d \}| $$ be the number of times that prime gap $d$ happens for primes less than $N$. Let $H = \...
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When is a number like “ddd…ddd”+1 (where d is a digit) a perfect square or a prime?

Inspired by Is the number $333{,}333{,}333{,}333{,}333{,}333{,}333{,}333{,}334$ a perfect square?, I wonder when numbers like these are perfect squares. Certainly, all numbers of the form $000...0001$ ...
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Understanding Newman's proof of the prime number theorem

I am trying to understand D.J. Newman's proof of the prime number theorem, as presented by D. Zagier. I am not too familiar with analysis, and so there are some things I don't understand. In (III), ...
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The divergence of the series of reciprocals of primes (proof check):

I want to check my attempt at a proof for the divergence of $$\sum_{n=1}^{\infty} \frac{1}{p_n} \tag{ $\star$ }.$$ We begin with assuming that $(\star)$ converges. If $(\star)$ converges, there is ...
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Sophie Germain Star prime number

Let us define following prime number : Let $~S_p~$ be Sophie Germain Star prime number of the form : $$S_p=12\cdot p \cdot (2p+1)+1,$$ where $p$ is a Sophie Germain prime . Note that ...
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Sum of odd prime and odd semiprime as sum of two odd primes?

How to prove that each sum of odd prime and odd semiprime can be written as sum of two odd primes $(p_1+p_2p_3=p_4+p_5)$ ? Since we know that each prime number greater than $3$ is of the form $6k\pm ...
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How can I speed up the search for a special number?

A number $N$ is given. The object is to find the smallest nonnegative integer $k$, such that $N+k$ is the product of three distinct primes, each having the same number of decimal digits. For example,...
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Reference request on the Riemann Zeta function

Let $\zeta^{-1}(x)$ be the functional inverse of the Riemann zeta function so that $$\zeta(s) = x \implies \zeta^{-1}(x) = s$$ The Riemann zeta function is injective in $1 < s < \infty$ where $...
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Is there a maximum number of consecutive sexy prime pair sums that are all divisible by $10$?

I was finding the sums of pairs of sexy primes (prime numbers that differ by 6) and noticed that there are a lot of pairs who's sum is divisible by $10$. Ex. $(7, 13)$ as $7+13=20$ and $20$ is ...
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Proof of Fermat primes and constructible n-gon

Prove that if a regular n-gon is constructible, then $n = 2^kp_1 ···p_r $ where $p_1,...,p_r$ are distinct Fermat primes using the following facts. If the regular $n$-gon is constructible and $n = qr$...
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Are 2, 3 the only prime numbers that don't have the digit 1 and are palindromes whose squares are also palindromes?

While thinking about prime numbers, I noticed that: $(1)$ Very few prime numbers have squares that are palindromes. Ex: $2$, $3$, $11$, $101$, $307$ $(2)$ Even rarer are prime numbers that are ...
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Why are some Ramanujan $G_n$ and $g_n$ functions highly factorable?

Given the Dedekind eta function $\eta(\tau)$ with $\tau = \sqrt{-n}$. Define the Ramanujan $G_n$ and $g_n$ functions as, $$G_n = 2^{-1/4}\frac{\eta^2(\tau)}{\eta(\tau/2)\,\eta(2\tau)}$$ $$g_n = 2^{-1/...
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Fractional parts of square roots of primes

To avoid confusion with other uses of braces, let $F:\Bbb R\to[0,1)$ be the fractional part function (usually noted as $\{\cdot\}$), so $F(x)=x-\lfloor x\rfloor$. It is known that the set $S:=\{F(\...
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Conjecture about primes and the greatest common divisor

Conjecture: Given $m,n\in\mathbb N^+$, one odd and one even, there are two primes $p,q$ such that $|mp-nq|=\gcd(m,n)$. I hope MSE can determine its validity. From time to time, when testing my ...
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Criteria for Leyland Numbers to be Prime

Inspired by a Black Pen Red Pen video on Youtube where he proved that $4^{2019}+2019^4$ is not prime using the Sophie Germain identity, I began exploring when numbers of the form $a^b+b^a;a,b\in\...
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About gaps between prime numbers

Consider some even integer number $n$. Let: $$C = \{c_i|i - n \mod i\}_{i=2..n}$$ For example for $n = 50$: $c_2 = 2 - 50 \mod 2 = 2$ $c_3 = 3 - 50 \mod 3 = 1$ $c_4 = 4 - 50 \mod 4 = 2$ $c_5 = 5 ...
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Can the Riemann Explicit Formula be used to find prime numbers?

It is well known that there is a strong link between the Riemann Hypothesis and the distribution of primes. The prime number theorem gives the number of primes less than or equal to a given $N$ as: ...
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Maximum runs of composites in arithmetic progressions

Is there a proof that every arithmetic progression of gap $p$ has a prime in the interval $[p, p^2)$? Put another way, can you prove the following: For all primes $p$, and all integers $0 \le m <...
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Does the arithmetic derivative have a geometric interpretation

The standard derivative, for example $\frac{d}{dx}x^2 = 2x, $ gives the slope of a function at a particular point. Does the arithmetic derivative have a similarly simple geometric interpretation? As ...
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Distributions of prime numbers

When folding the number line not into a spiral (like Ulam did) but into a zig-zag pattern (like Cantor did) there are other patterns visible in the distribution of prime numbers: [To see these ...
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The hypothesis of the unbounded product of the digits of a prime number

Hypothesis: the product of the digits of prime numbers is not bounded from above This task arose from one research task, so maybe it is very complex and I do not know how to approach it. I will ...
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Is there a name for this “Collatz constant”?

Right now I'm calling a convergent number based on the Collatz conjecture the "Collatz constant". I'm wondering if it have an ACTUAL name? And if it actually converges? Details The Collatz ...
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Can $2^s+3^s+\cdots +q^s$ , $q$ odd prime , $s\ge 2$ be a perfect power?

Let $q$ be an odd prime and $s\ge 2$ be an integer. Define the sum $$S(q,s):=\sum_{p\ prime,p\le q} p^s=2^s+3^s+\cdots +q^s$$ Can $S(q,s)$ be a perfect power ? Among other searches, I searched for ...
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Probability not divisible by $k^2$

My question is about this problem: Let $s > 1$ and $$\zeta(s) = \sum_{n = 1}^{\infty} n^{-s}.$$ Furthermore let $(\Omega, \mathcal{F}, \mathbb{P}$) be a probability space with $\Omega = \mathbb{N}$...
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Is the occurence of two perfect numbers a coincidence?

The expression $$n^n+n!+1$$ is prime for the following non-negative integers $n\le 7\ 600$ : $$[0, 1, 2, 4, 28, 496]$$ if we assume $0^0=1$ The numbers $28$ and $496$ are perfect numbers. Is ...
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An approximation for $1\leq n\leq N$ of the number of solutions of $2^{\pi(n)}\equiv 1\text{ mod }n$, where $\pi(x)$ is the prime-counting function

We denote the prime-counting function with $\pi(x)$ and we consider integer solutions $n\geq 1$ of the congruence $$2^{\pi(n)}\equiv 1\text{ mod }n.\tag{1}$$ Then the sequence of solutions starts as $$...
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Are there more non-trivial powers of the form $S_p$?

Let $S_p:=2+3+5+\cdots p$ denote the sum of primes up to $p$. I searched for $S_p$ being a non-trivial power and only found the following squares so far : ...
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Prime candidates of the form $n^{(n^n)}+n^n+1$?

Let $$\large f(n)=n^{(n^n)}+n^n+1$$ Checking $f(n)$ for $2\le n\le 100$, I noticed that $f(n)$ has a small prime factor except for $n=12,53$ and $60$ For $n=53$, I found the prime factor $7074407$ , ...
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Erdős-Kac and moments

One way to prove Erdős-Kac is by proving that for each non-negative integer $k$, the $k$-th moment $$M_k(x):= \frac{1}{x}\sum_{1 \leq n \leq x} \left(\frac{\omega(n)-\log \log x}{\sqrt{\log \log x}}\...
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The Miller-Rabin Test clarification

I'm teaching myself cryptography but have realised that it has a lot of number theory as a part of it, one area which I'm a bit confused over is the Miller-Rabin test and how to use it in questions. A ...
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84 views

Simplify a prime testing function

A while ago, I came up with this formula: $$\sum_{n=2}^{\lfloor\sqrt{x}\rfloor}\left\lfloor\frac{x}{n}-n+1\right\rfloor$$ Which, for the given value of $x$, returns the number of factor pairs that ...
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prove $p|\left(2+\sqrt{5}\right)^p-2^{p+1}$

Can anyone help me? I know that first part is never an integer and second is, so how is possible that the number which is not an integer is divisible by an integer? p is prime number and p>2
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Penrose tilings and prime numbers

Can one exclude the idea of prime numbers being a projection (using a proper acceptance domain) of a higher dimensional lattice onto 1D similar to the generation of Penrose tilings (see e.g. thesis)? ...
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How many primes of the form $2^p-p$ with $p$ prime?

I know that if $p = 6k+1$ with $k$ integer, then $(2^p-p) \mod 6 \equiv 1$. I think that this means that $2^p-p$ could be prime. My question is: are there a finite or infinite number of primes which ...
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Claim: $a$ has $90 \% $ primes less than $n$ If $n!= 2^s \times a \times b $ and $\lfloor{\frac{a}{b}}\rfloor = 2^{s-2}$

Description: We can write, $n!= 2^s \times a \times b \cdots (1)$ where $gcd(a,b)=1$ and $2^{s+1} \nmid n!$ .It is given that, $\lfloor{\frac{a}{b}}\rfloor = 2^{s-2}$. Claim: If $n!= 2^s \...
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Is it true that the probability that an “almost prime” is prime, tends to $0.59$?

According to the estimation of the number of rough numbers I found in Wikipedia, the number of integers from $1$ to $x$ with no prime factor below $x^{1/u}$ tends to $\omega(u)\frac{x}{\ln(x^{1/u})}$ ,...
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Is the error I noticed a harmless typo?

Here http://arxiv.org/PS_cache/arxiv/pdf/1002/1002.0442v1.pdf , at page $2$ at the bottom, it is stated that the number of primes not exceeding $x$, denoted by $\pi(x)$, satisfies the double-...
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Do DSm(n) never be a prime?

Smarandache number (denoted as Sm(n)), is a number formed from concatenating the first n natural numbers. For example Sm(11)= $1234567891011$ (http://mathworld.wolfram.com/SmarandacheNumber.html). And ...
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What percentage of prime number factorials plus 1 are themselves prime?

One of the steps in Euclid's proof of the infinity of primes is sometimes misinterpreted to be a way of generating new prime numbers. Specifically constructing the number P!+1 where P is a prime is ...
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How close are we to knowing the rate of convergence to $0$ of $\prod_{p\le x}(1-1/p)^{-1}-e^\gamma\log x $?

This is a question related to an earlier one of mine, which I may answer myself eventually, as I have learnt more about the topic. Despite what one can read on the MathWorld page about Mertens' third ...
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If $\frac{1}{8} \left(5^m-2\cdot 3^m+1\right)$ is prime, then $m=2p$ where $p$ is prime?

The following statements are easy to prove with elementary arguments: $X_m=\frac{1}{8} \left(5^m-2\cdot3^m+1\right)$ is an integer for all integers $m\ge 0$ ($m \equiv 0 \mod 4$ or $m \equiv 1 \...
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Perfect Prime $4D$-Cube

$\color{gray}{\mbox{ I don't want to overflow/burden you with numerous rigorous definitions...}}$ So, definition by the images:      Perfect Prime Cube:   $(30; 5, 22, 78)^...
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Power series with products of prime numbers as coefficients

I encountered in my work an infinite series (only even power) with $n^{th}$ coefficient being of the following form: $$\frac{1}{3\cdot5\cdot7\cdot11\cdot...\cdot(n^{th} prime)}.$$ I wonder if such ...
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How big are the factors of $2^kp - 1$?

Let $p$ be a prime number, $p > 3$. Does it always exists a $k \in \mathbb N, k \ge 1$ such that the prime factors of $2^kp - 1$ are all less then $p$? Thoughts Well we can easily see that ...
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Approximate zeros of a (hypothetical) analog of $\zeta(s)$

[Added numbers 11/13.] Motivation (can skip). When prime powers $p_n$ are used to calculate $$y(x) = \sum_{n=1}^{N}\frac{\sin (x \log p_n)}{p_n},\hspace{5mm}(1)$$ for (say) $N= 30,$ $x>5$, at ...
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Is there a quicker way to generate integers which are hard to factor than multiplying two large primes?

An easy way to generate an integer which is hard to factor is to find two large primes and multiply them. As a bonus, you know the factors. I'm interested in whether it's possible to find integers (...
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Number of lucky primes

The "lucky numbers" can be constructed with this sieve. The red ones are the lucky numbers. As you can see, some are prime. Is the number of lucky primes infinite? Edit: Apparently this is an open ...
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How prove $\sigma(4^p-1)<(2^{p+1}-1)^2$

If $p$ is an odd prime numbers, show that $$\sigma(4^p-1)<(2^{p+1}-1)^2$$ where $\sigma(n)$ stands for the sum of divisors. I thought of using the formula for $\sigma(n)$: If $4^p-1=3^k\cdot (...
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A question on the Pell equation $x^2-pqy^2 = -1$, with prime $p,q$.

We know that a necessary but not sufficient condition such that, $$x^2-dy^2 = -1\tag1$$ is solvable is that $d$ is not divisible by a prime of form $4m+3$. It is not sufficient because the prime ...
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Prove that $E_0$ is transcendental

Consider the non-negative natural numbers: $0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19\dots$ Encode the primes as $1$, the rest as $0$. $E = 0,0,1,1,0,1,0,1,0,0,0,1,0,1,0,0,0,1,0,1\dots$ ...