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.

learn more… | top users | synonyms

2
votes
0answers
33 views

What is a good tool for this job involving the prime spiral?

I'm interested in studying the prime spiral interactively. This question talks about some interesting patterns in the spiral involving quadratic equations. The idea I had was, write a program that ...
2
votes
0answers
73 views

Prime numbers of the form: $k\cdot 2^n \pm 1$ , where $k<3n$

Is it true that : For every $n$ there exists a number $k<3n$ such that: $k\cdot 2^n-1$ or $k\cdot 2^n+1$ is prime,where $k,n\in \mathbf{N}$ Maple code that prints least $k$ such that ...
2
votes
0answers
68 views

Prime numbers of the form : $2^{n+a}+2^{n} \pm 1$ , where $0 \leq a < n$ and $n \equiv 0 \pmod 6$

Is it true that : For any positive integer $n$ such that $n \equiv 0 \pmod 6$ there is at least one prime number of the form: $p=2^{n+a}+2^{n} + 1$ , or , $p=2^{n+a}+2^{n} - 1$ with ...
2
votes
0answers
164 views

$n$ by $n$ Primally Magic Squares

(Again copied verbatim from a September 2009 thread I made.) A Primally Magic Square (PMS) is exactly like a traditional magic square with a change of criteria. Where a traditional magic square is ...
1
vote
0answers
21 views

Question on Newman's proof of the Prime Number Theorem

I am reading through Zagier's exposition of Newman's proof of the prime number theorem and I do not understand one of his arguments when proving his so called Analytic Theorem. This theorem states the ...
1
vote
0answers
34 views

Convergence of recurrence relation involving divisors

I$\let\leq\leqslant\let\geq\geqslant$ thought up a family of sequences, recursively defined by $$a_{n+1}=\frac{d_n^ra_n+a_{d_n}}{d_n^r+1}\quad(n\geq2)$$ where $r,a_1,a_2\in\mathbb R$ are parameters ...
1
vote
0answers
27 views

When looking at the mod as binary value

Look at the next value: $$617*947 = 584299$$ 617, 947 are prime values. I want to see what are all the possible solutions for the next equation, for $k=4$: $$(a\mod k)(b\mod k) = 584299\mod k$$ ...
1
vote
0answers
65 views

Reasoning about $z^n = x^m + y^m$

Let $z,n,x,y,m$ be positive integers with $z \ge 5$ and $m \ge 3$ and $m$ odd. Does it follow that: $z$ cannot be prime if $p \ge 5$ and $p | z$, then either $p > m$ or $p|m$ Here is my ...
1
vote
0answers
64 views

Can this approach to showing no positive integer solutions to $p^n = x^3 + y^3$ be generalized?

The following problem is a $2000$ Hungarian Olympiad question. Find all primes $p$ such that: $$p^n = x^3 + y^3$$ The answer is that there are only $2$ solutions: $2^1 = 1^3 + 1^3$ $3^2 = 2^3 + ...
1
vote
0answers
33 views

Lucas-Lehmer primality test and numbers form $k^{n}-1$

Is it possible to use Lucas-Lehmer test for testing the primality of numbers of the form $k^{n}-1$, where $k > 2$? For $k = 2$ (Mersenne number) $c_0 = 4$. What would be $c_0$ for $k > 2$?
1
vote
0answers
34 views

Bunyakovsky conjecture for cyclotomic polynomials

This article on Wikipedia: http://en.wikipedia.org/wiki/Bunyakovsky_conjecture says: In fact, it can be shown that if for all natural number $ n $, there exists a natural number $ x > 1 $ such ...
1
vote
0answers
34 views

Is there a better way to search a string so that I don't get false positives?

I've been using this equation to find primes of a specified range, in this example the range is 10 to 10000. Go ahead and try it out, your monitor will not catch fire. The problem I'm now having is ...
1
vote
0answers
82 views

Is every integer $\geq5$ the sum of two primes and a power of a prime?

Is every integer $\geq5$ the sum of two primes and a power of a prime (where $1$ is included in the prime powers)? I don't really expect someone to prove this here, but I wonder if the question has ...
1
vote
0answers
31 views

Solutions $n^2 = -1 \mod (p_n-1)$

Consider the equation $n^2 = -1 \mod (p_n-1)(*)$ where $p_n > n$ and $f(n) = p_n$ is the largest prime that satisfies the equation. $f(n)$ gives $p_n$ assuming there is a solution to the equation ...
1
vote
0answers
40 views

Has anyone established an upper bound for the least integer $k$ such that infinitely many primes have at most $k$ ones in their binary representation?

Has anyone established an upper bound for the least integer $k$ such that infinitely many primes have at most $k$ ones in their binary representation? $2$ is the only prime with $1$ one, the Fermat ...
1
vote
0answers
60 views

Riemann's explicit formula for $\pi(x)$

Riemann's explicit formula $J(x)=\mathrm{Li}(x)-\sum_{\Im\varrho>0}\left(\mathrm{Li}(x^\varrho)+\mathrm{Li}(x^{1-\varrho})\right)+\int_x^\infty\frac{\mathrm{d}t}{t(t^2-1)\log t}-\log2,$ where ...
1
vote
0answers
268 views

What is the next such prime

This is about primes with "very" interesting forms. Such this one: primes $p$ such that the concatenation of first $k$ primes with only prime digits (i.e. $2$, $3$, $5$ and $7$) from $2$ to $p$ is a ...
1
vote
0answers
33 views

On integer $n>1$ and prime $p$ such that $p<n$ , $p$ does not divide $n$ and $n-p$ is a prime

Let $n>1$ be a given integer and $p$ be a prime less than $n$ and not dividing $n$ ; so $p$ and $n$ are co-prime ; hence $n-p$ and $n$ are also co-prime ; I would like to ask when is $n-p$ also is ...
1
vote
0answers
26 views

How quickly can we multiply hypercomplexes?

If we start with a hypercomplex number with $2^n$ entries, how quickly can we multiply it by another hypercomplex number, modulo a prime? EXAMPLE For example, with $n=1$, we get the complex numbers. ...
1
vote
0answers
36 views

Numbers with special factorisation

We know that any natural number $n$ can be decomposed as $p_1^{k_1}p_2^{k_2}...p_n^{k_n}$. I am looking for numbers which have $k_1=k_2=k_3=....=k_n=1$ i.e. given a number n, identify if it has all ...
1
vote
0answers
37 views

prime case function?

Does there exist a name (or assigned to a mathemtician) for a case function $f(x)$ in literature, such that it twould take the value $1$ when $x$ primes, and zero otherwise? I am just looking for a ...
1
vote
0answers
27 views

Composite residuosity statement.

Consider the following definition. A number $z$ is said to be $n$-th residue modulo $n^2$ , if there exists a number $y \in \mathbb{Z}_{n^2}^*$ such that $$z\equiv y^n \mod n^2$$ Let us take $n=6$ ...
1
vote
0answers
55 views

How many unique combinations of sets can we get?

Starting with $x$, which is a positive integer or zero, and $y$ a second positive integer or zero, with $y \ne x$, we can create lists. Set $p$ a prime greater than 2, $\alpha = (p-1)/2$, and ...
1
vote
0answers
62 views

Golbach's partitions: is there always one common prime in G(n) and G(n+6) , n greater or equal to 8 (or a counterexample)?

I am trying to find a counterexample for the following expression when d=6. (G(n) = Goldbach partition of the even number n) ${\forall}$ n=2*k / k${\in}$N, n${\geq}$8 ${\exists}$(${p_i}$,${p_j}$) / ...
1
vote
0answers
51 views

Proving multiplicative property of euler's totient function $\phi$ using probability

If $m,n$ are co-prime , we know that $\phi(mn)=\phi(m)\phi(n)$. I want to prove it using probability. Probability that a selected number less than or equal to $mn$ is co-prime to $mn$ = ...
1
vote
0answers
35 views

Group, QR, QNR, Product of distinct primes

$N = pq$ where $p$ and $q$ are distinct primes. $ZN^*$ is all $x$ belonging to $ZN$ such that $gcd(x, N) = 1$. How do I find if $ZN^*$ is closed under addition? I believe $QR \times QR$ gives a ...
1
vote
0answers
79 views

Primality of Stirling numbers of second kind (again)

This question follows a previous one on the primality of Stirling numbers of the second kind ${n \brace k}$. Gerry indicated a paper on the topic. In this paper it is shown that for ${n \brace k}$ to ...
1
vote
0answers
23 views

How to split a list in n parts so the calculation time will be equal

I'm trying to implement a prime number finder. It as to find primes from 0 to X. I use this algorithm (performance may be questionable but this is not the question) to find the primes : ...
1
vote
0answers
25 views

Syndeticity and A.P.-richness of certain sets

Let $A \subset \mathbb{N}: \sum_{a \in A} (\frac{1}{a}) = \infty$; denote $\{ \alpha_1 @ \alpha_2: \alpha_1, \alpha_2 \in A \} = A @ A$, where "$@$" is any appropriate binary operator. (Note: $A$ is ...
1
vote
0answers
35 views

Fundamental Theorem of Arithmetic (Canonical) missing crucial step

I've worked long on the proof of the fundamental theorem of Arithmetic and there is only one tiny piece left I can't wrap my head around. Suppose that $$\prod_{i=1}^r p_i^{m_i} = \prod_{j=1}^s ...
1
vote
0answers
156 views

Remarks on a Previous Post

Recently I have been reading this post and I have noted something significant in the fake argument. As one can easily see that the basic idea behind the argument had been to show that the sequence ...
1
vote
0answers
48 views

The Existence of “Simple” Prime Generating Functions

Obviously, we do not know an explicit and easily manipulable formula for finding every prime - that is, a function $f(n)$ which yields the $n^{th}$ prime. I've seen plenty of formulas that "cheat" in ...
1
vote
0answers
35 views

Questions about primes made from consecutive numbers starting from 1

Similar to: Does there exist a prime that is only consecutive digits starting from 1? Let $b_n=\overline{a_1a_2a_3\dots a_n}$ and $a_n=n$. For example $b_{11}= 1234567891011$. I have a couple of ...
1
vote
0answers
52 views

Proof concerning specific class of Proth numbers

Is this proof acceptable ? Theorem Let $N = k\cdot 2^n+1$ with $n>1$ , $k<2^n$ , $k$ odd and $3 \nmid k $ , thus $N$ is prime iff $3^{\frac{N-1}{2}} \equiv -1 \pmod N$ Proof Necessity ...
1
vote
0answers
49 views

Prime numbers that fits in a specific pattern

Any series $\displaystyle \sum_{k=0}^{\infty}a_k2^{-k}$, where $a_k\in\{0,1\}$, converges to some $x\in[0,2]$ and since the sequence $a_n$ is unique for each $x\in[0,2]$ there is an bijection between ...
1
vote
0answers
30 views

Could you give a definition of what is a superior highly composite number using only words?

I know very well what is a superior highly composite number, but I would like to see how could we (roughly) define what is a superior highly composite number using only words (using no equations and ...
1
vote
0answers
46 views

Difference between two (not consecutive) primes

I am searching for an lower bound on the difference between the $(n+k)$-th and $n$-th prime number in terms of $k$. I have something like this in mind (conjecture): Let $(p_k)_k$ denote the ...
1
vote
0answers
48 views

Sequence of primes by concatenating digits in a given base.

Given a base, $b$ is there is a sequence $\lbrace a_n\rbrace_{n\geq 0}$ where $a_k \in \lbrace 1,2\cdots, b-1\rbrace$so that the sequence: $$b_n:= \sum_{k=0}^n a_kb^k$$ is a sequence of primes ...
1
vote
0answers
58 views

Fast algorithm for generating consecutive primes larger than N

I'm looking for a fast algorithm to generate primes larger than a random 4096 bit number $N$. I know about the Sieve of Atkin, but AFAIK it can only be used to find all primes up to a certain limit. ...
1
vote
0answers
31 views

Cramér's Model - “The Prime Numbers and Their Distribution” - Part 3

Following a previous question (here you'll find an introduction): The book states that almost surely $$\pi_S(x+y)-\pi_S(x)=\mathrm{li}(x+y)-\mathrm{li}(x)+O(\sqrt y)$$ as soon as $y/(\log ...
1
vote
0answers
43 views

A Question Related to Zsigmondy's Theorem

I am wondering if there is a way to prove the following statement, which bears some resemblance to Zsigmondy's Theorem. I am not sure if the statement is true, but it seems as though it should be. ...
1
vote
0answers
91 views

How I could transform this into product over primes :$s_p$= $\frac{1}{2^2-1}+\frac{1}{2^3-1}+…\frac{1}{2^p-1}$?

1)Can I transforme this sum into product OVER primes:$s_p$= $\frac{1}{2^2-1}+\frac{1}{2^3-1}+....\frac{1}{2^p-1}$ ? Note : p is prime number and ${2^p-1}$ is prime 2)I would be interest to know ...
1
vote
0answers
39 views

RSA aloghorithm - stuck on d

I'm sorry in advance if this sort of question has been posted before. I couldn't find it. I'm clearly an idiot, and I clearly need help, so here I am. I have a homework assignment which overall is ...
1
vote
0answers
55 views

Wolstenholme Number

Does Wolstenholme Numbers have perfect squares other than 1 and 49? The first few are 1, 5, 49, 205, 5269, 5369, 266681, 1077749 seems to be a complicated problem
1
vote
0answers
34 views

On non-divising primes of an integer $x$

We know more about divisor than non-divisors, If we consider the sets : $$P^{1}_{x} =\left \{ p \leq x : \ p \in \mathbb{P} \right \}$$ ($\mathbb{P} $ is the primes set) $$ P^{2}_{x} =\left \{ ...
1
vote
0answers
41 views

What is the status of research on primes as an example of general sieve-generated sequences?

I have been interested in treating the prime numbers as a special case of sieve-generated sequences, however they may be defined by different authors. Can someone here give me any information about ...
1
vote
0answers
32 views

Is there a Poulet number with this condition?

Is there a Poulet number $n$ with this condition: $◎(n)=\frac{n+1}{2^x}$ or $\ ◎(n)=\frac{n-1}{2^x}, \ x \in \mathbb{N}_{\gt 0}$? (Recall that a Poulet number is a composite $n$ such that $2^n−2$ is ...
1
vote
0answers
41 views

Existence of primes $p$ such that all the prime divisors of $p+1$ divide $p-1$

This question recently came up to me in a project and is not taken from a textbook. I would like to know if any characterization of such primes is known from literature. They are seemingly rare but do ...
1
vote
0answers
47 views

Are there such prime giving functions?

Here let us define a function $f : \mathbb{N} \rightarrow \mathbb{N}$ , such that for every $n$ , The sequence $\{f(n) ,f(n)+1 ,f(n)+2 , f(n)+3, \dots , f(n)+n\}$ contains atleast $1$ prime . Let us ...
1
vote
0answers
69 views

Prime Zeta Function

Does $$\sum_{p \text{ prime}} \frac{1}{p^s} \sim \log \zeta(s) \quad \text{as} \quad s \to 1^+$$ imply $$\sum_{p \leq n} \frac{1}{p} \sim \log H_n \quad \text{as} \quad n \to \infty,$$ where $H_n$ is ...