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

1
vote
2answers
122 views

Mill's Constant Unpublished Extras

The brown paper used in the making of Numberphile's video on Mill's Constant was recently sold on eBay. Here is an image of it, from the eBay listing, The bottom two lines appear on the brown ...
1
vote
1answer
86 views

Relation between $1-(n^{p-1}\mod p)$ and Riemann $\zeta$

Taking: $$\mathcal V_p=1-(n^{p-1}\mod p)$$ with $$\lim_{p\rightarrow \infty}\mathcal V_p = \operatorname{sinc}(2\pi \,n)$$ and Riemanns' well known functional equation, I get easily to this result: ...
1
vote
2answers
265 views

“Interesting” Sequences

Well, here's a question i myself made up and i thought it's interesting if i share it with everyone. We call a sequence of natural numbers (for example $a$) Interesting if (all three must be true): ...
3
votes
2answers
119 views

How many squares are there modulo a Mersenne prime?

Mersenne primes are primes of the form $M_n = 2^n - 1$. I'm wondering how many distinct natural numbers result from squaring the naturals modulo $M_n$. As an example, $M_3 = 7$. If we take the ...
3
votes
1answer
101 views

What do we know about the distribution of Mersenne primes?

Mersenne primes are primes of the form $M_n = 2^n - 1$. I'm wondering how far apart successive Mersenne primes can be. For example, is $M_{n+1} \le O((M_n)^e)$? Or, is $M_{n+1}$ always less than ...
1
vote
6answers
2k views

“Question: Show that $n^5 - n$ is divisible by 30; for all natural n” [duplicate]

Show that $n^5 - n$ is divisible by $30;$ $\forall n\in \mathbb{N}$ I tried to solve this three-way. And all stopped at some point. I) By induction: testing for $0$, $1$ and $2$ It is clearly ...
2
votes
1answer
84 views

Counting the number of distinct integers in a range that fit a specified pattern

I've been thinking about primorials in the context of the twin prime conjecture. I am seeing this primarily as an exercise to improve my intuition about primorials and prime patterns more than the ...
15
votes
2answers
519 views

Is there a better upper bound for the primorial $x\#$ than $4^x$

In the classic proof of Bertrand's postulate by Paul Erdős, he shows that $x\# < 4^x$ where $x\#$ is the primorial for $x$. Is there any tighter upper bound for a given primorial $x\#$? Ideally, ...
1
vote
1answer
200 views

Littlewood's 1914 proof relating to Skewes' number

From Littlewood's 1914 theorem (paraphrase): I propose to show there are arbitrarily large values of x for which successively $\psi(x) - x < - K\sqrt{x}\log\log\log x \tag{A}$ $ ...
8
votes
4answers
284 views

Gaps between primes

I recently watched a video about the recent breakthrough involving the gaps between primes. I have an idea that I'm sure is wrong, but I don't know why. If you take the product of all prime numbers ...
0
votes
1answer
575 views

Extension of Fermat's little theorem with Carmichael numbers

I'm a bit confused about the nature of one of my homework problems. It is requesting an explanation for why a congruence holds for $a^n \equiv a \;(\!\!\!\mod n)$ for a composite $n$, however this ...
1
vote
2answers
60 views

Number Theory congruence classes

if $n=p^2$ ($p$ is prime) if $[x]=[1]\mod p$, Then What is $[x]$ in$\mod n$? i.e. $[x]=[?]\mod n$ where [x] belongs to (Zn)* where (Zn)* = {[x] belonging to Zn such that gcd(x.n)=1)
6
votes
1answer
242 views

A Conjecture about Maximal Prime Gaps

As it is well known that prime number is $2,3,5\cdots \cdots$, thus all these prime number are denoted by$p_{1},p_{2},\cdots \cdots ,p_{n}\cdots \cdots$. The prime maximal gap $\max_{p_{n+1}\leqslant ...
0
votes
1answer
114 views

Are there any algorithms to check if a big number is a prime number?

I want to check if a given number is a prime number. Are there other ways than brute-force? It should be fast and work with bigger numbers (>1.000.000).
8
votes
0answers
111 views

Asymptotics of the lower approximation of a pair of natural numbers by a coprime pair

When we are working, for instance, in combinatorics or graph theory, sometimes we can have the following situation. For each number $m$ from an infinite set $\mathbb M\subset\mathbb N$ we can ...
1
vote
0answers
80 views

Deriving this recursive expression for Riemann Prime Counting Function?

Why does this work? $f(n,k,1)=0$ $f(n,k,j)= \frac{1}{k} - f(\lfloor\frac{n}{j}\rfloor, k+1, \lfloor\frac{n}{j}\rfloor) + f(n,k,j-1)$ Here, f(n,1,n) computes the Riemann Prime Counting Function. ...
2
votes
1answer
148 views

Mill's formula, $\theta^{3^n}$ is a prime for a certain $\theta$ and all natural $n$?

I just watched this video done by numberphile, and the video claimed that there exist certain numbers $\theta$ such that the floor function of $\theta^{3^n}$ is a prime for all natural $n$ . $\theta$ ...
2
votes
2answers
63 views

Avoiding primefactors in reducible polynomials

Take distinct pairs $(c_i,d_i) \in \mathbb Z^2$, the entries being coprime. Put $f(x) = \prod_{i=1}^k (c_i x + d_i)$. Let $\mathbb P$ denote the rational prime numbers. Which conditions (if any) need ...
1
vote
1answer
51 views

Closed forms for $\lim_{x\rightarrow \infty} \ln(x) \prod_{x>(p-a)>0}(1-(p-a)^{-1})$

Im looking for closed forms for $\lim_{x \rightarrow \infty} \ln(x) \prod_{x>(p-a)>0}(1-(p-a)^{-1})$ where $x$ is a positive real, $a$ is a given real, $p$ is the set of primes such that the ...
5
votes
1answer
81 views

Where are the zeros of $\prod\limits_p (1-(p-1)^z)$?

Define $f(z)$ as the analytic continuation of $\prod\limits_p (1-(p-1)^z)$ where $z$ is complex and the product is over the odd primes $p$. Where are the zeros ($f(z)=0$) of this function ?
0
votes
1answer
77 views

To which extent distribution of Riemann non-trivial zeros follow a gauss process?

I am trying to clearer and preciser understand to which extent the distribution of the non-trivial zeros of the Riemann $\zeta$-function follow a Gauss process? Yet, what I figured out from ...
1
vote
2answers
164 views

Bound on number of zeros in smallest prime greater than $10^n$

I'm inspired by this comment by Eugene Wallingford on a blog post by John D. Cook. Take this Python code, using SymPy (note, I used this branch to ensure that ...
3
votes
1answer
123 views

A curious fact on partitions of 7 integer and related question.

Let's start writing $7$ partitions, marking them with $n\ A$, where $n$ is a number of terms in partition and $A$ is a set of terms in it. $$\underbrace {7}_{1\ \{7\}} = 7$$ $$\underbrace {6 + ...
7
votes
5answers
576 views

Why it is important to find largest prime numbers?

It always takes a lot of effort and money to find the next largest prime number. Why is it so important to do this work and what is the application those numbers?
0
votes
1answer
84 views

Discrepancy between terms of sum and sum

My question is why the following happens, and whether we can correct (2) below to account for an errant factor of 2. By a slight generalization* of the argument of this problem we have I think that ...
5
votes
2answers
229 views

Primes and the Unit circle.

Consider the "prime spiral" $f(z) = \sqrt{z}\exp(2\pi i \sqrt{z})$, for integer $z$. It has been shown that the intersections of $f$ with some quadratic curves contain a significantly disproportionate ...
5
votes
4answers
394 views

Solutions to $p+1=2n^2$ and $p^2+1=2m^2$ in Natural numbers.

$$p+1=2n^2$$$$p^2+1=2m^2$$ Find positive integers $m,n$ and prime $p$ satisfying the above two equations. What would people commonly do? Subtracting both the equations. You get: ...
1
vote
1answer
92 views

Series equivalent to $\sum p_k$

Looking at a theorem of Chebyshev, I noticed that $$\sum_{n=0}^{\infty} \sum_{p_k < n} \frac{(\log p_k)^n}{n!} = 2 + 3 + ...+ p_k.$$ Proof. Letting $x = \log p_k$ and writing out the expansion of ...
4
votes
2answers
263 views

What is the need for classifying numbers like integer, whole number etc?

what are the everyday life examples where we use the classification. I feel all the math behind the scenes(in computers weather etc ) is highly abstracted. I am looking for strong answers to tell the ...
0
votes
3answers
658 views

Prove that every integer is either prime or composite

In the book I'm reading, the following proof is given for the stated theorem: Let n be any integer that is greater than 1. Consider all pairs of positive integers $r$ and $s$ such that $n = rs$. ...
0
votes
1answer
192 views

Infinitely many primes of the type 5 mod 6.

Problem: Prove that there are infinitely many primes of the type 5 mod 6. My professor did the problem and the proof was horribly long. Can someone show me a shorter version of the proof of this ...
4
votes
2answers
150 views

Limiting value of $\lim \frac{1}{k}\sum_{n=1}^k \frac{p(n+1)-p(n)}{\log p(n)}$

Empirically it seems $$\lim_{k\to \infty} \frac{1}{k}\sum_{n=1}^k \frac{g(n)}{\log p(n)} = 1\tag{1} $$ in which p(n) is the nth prime and g(n) is the prime gap $p(n+1)-p(n).$ Cramer conjectured ...
1
vote
2answers
109 views

Does $f(n)\sim g(n)$ imply $\lim_{k\to\infty} \frac{1}{k} \sum_n f(n)/g(n) = 1$?

Is it true that $$\lim_{k\to\infty}\frac{1}{k}\sum_{n=1}^k \frac{f(n)}{g(n)} = 1 \leftrightarrow f(n)\sim g(n).$$ My thought: $f(n)\sim g(n) \rightarrow \frac{1}{k}\sum \frac{f(n)}{g(n)} = 1$ since ...
2
votes
1answer
167 views

Limit of $\sum\frac{1}{p(\pi(n))}$

Let $p(n)$ be the nth prime and $\pi(n)$ the number of primes not exceeding n. I wonder if we can show that $$\tag{1} S = \sum_{n= 2}^k \frac{1}{ p (\pi (n))} \sim \log k. $$ We know by comparison ...
12
votes
5answers
3k views

Why does one counterexample disprove a conjecture?

Can't a conjecture be correct about most solutions except maybe a family of solutions? For example, a few centuries ago it was widely believed that $2^{2^n}+1$ is a prime number for any $n$ . For ...
1
vote
0answers
104 views

Lower bound for $\pi(x)$

Is there a way to show that $$\frac{x}{\ln x} < \pi(x),$$ for sufficiently large $x$, using only elementary calculus? Apparently it is true for $x \geq 17$ (see this article). However, I am looking ...
2
votes
1answer
156 views

There is a real number $\alpha >1$ such that $\Bigl\lfloor2^{2^{{.}^{{.}^{{.}^{2^{\alpha }}}}}}\Bigr\rfloor$ is prime for all $n\geq 1$

Theorem: There exists a real number $\alpha >1$ that if $$\alpha =\alpha _0,\quad 2^{\alpha _0}=\alpha _1,\quad \dots\quad 2^{\alpha _n}=\alpha _{n+1},\quad \dots$$ then for all $n\geq ...
0
votes
1answer
393 views

Quadratic expression that generate primes

I recently learned that there exist quadratic expression that generate some primes and some of these equations generate more primes than others. In the following video, the person shows the following ...
7
votes
1answer
224 views

Prime numbers in Collatz sequences

This question/request is twofold. First, if this is a stupid question or if it has been addressed before, please say so (bluntness is optional), and I will crawl back into my cave... My question: is ...
7
votes
3answers
269 views

The ordinary generating function for $ζ(s)$

$$\zeta(s)^m = \sum_{n=1}^{\infty} \frac{a_n}{n^s}$$ where $ζ(s)$ is the Riemann zeta function has the ordinary generating function: $$\sum \limits_{n=1}^{\infty} a_nx^n = x + {m \choose 1}\sum ...
2
votes
1answer
1k views

Sum of the first n Prime numbers

Let $P_i$ denote the i-th prime number. Is there any formula for expressing $$S= \sum_{i=1}^m P_i.$$ We know that there are around $\frac{P_m}{\ln(P_m)}$ prime numbers less than or equal to $P_m$. ...
13
votes
2answers
247 views

Prove that $m^{2013}-m^{20}+m^{13}-2013$ has at least $N$ prime divisors

for positive integer $N>1$,There always exists $m$ such that $$m^{2013}-m^{20}+m^{13}-2013$$ has at least $N$ prime divisors Thank you all, this is good problem, but I don't know how to solve it.
2
votes
2answers
81 views

A function that brings back the prime number just before it?

Is there a function that brings the prime number just before it? I.e P(18)=17 P(6)=5 P(28)=23; I know how weird that sounds.
-1
votes
3answers
508 views

What are the connections between number theory and topology ?? [closed]

What are the connections between number theory and topology ? How does topology relate to number theory ? In particular I wonder about primes and diophantine equations. I do not see how the amount of ...
13
votes
2answers
327 views

Number of digits until a prime is reached

Begin with a random digit from $1$ to $9$. Add a random digit to the right-hand side from $0$ to $9$ until a prime number is reached. How many digits are necessary in the avarage ? More precisely: ...
7
votes
1answer
147 views

How can prove this $\binom{n}{p}\equiv \left\lfloor\frac{n}{p}\right\rfloor \pmod {p^2}$

Show that if $n \gt p \gt 0$: $$\binom{n}{p}\equiv \left\lfloor\dfrac{n}{p} \right\rfloor\pmod{ p^2}$$ where $p$ is prime. and $$\binom{n}{p}=\dfrac{n!}{(n-p)!p!}$$ This is theorem? True or false? If ...
3
votes
0answers
90 views

An intuitive interpretation of Montgomery pair corrlation function vs. prime divisibility?

Theorem - If the Riemann hypothesis would be true, and the Montgomery pair correlation conjecture (see linked article page 183-184) true too; let $p \in \Bbb P$ prime, $n \in \Bbb N$ and $$\mathcal ...
-1
votes
1answer
73 views

Conjecture on limit of $1-(n^{p-1}\mod p)$

Given $p \in \Bbb P$ prime, $n \in \Bbb N$ and $$\mathcal V_p=1-(n^{p-1}\mod p)$$ let me conjecture that $$\lim_{p\rightarrow \infty}\mathcal V_p = \operatorname{sinc}(2\pi \,n)$$ Question: Is ...
2
votes
0answers
198 views

Prime number finding via polynomials

I try to find approximation polynomial to estimate which number is prime or not. Addtion to this, (If It is possible) To find the closed form of coefficients of the series ($c(n)$) Euler found the ...
4
votes
1answer
200 views

Length of recurrent strings of numbers in the decimal expansion of $1/p$, where $p$ is prime.

Am I right to assume that: all rational numbers have a recurrent sequence in their decimal expansion, and the length of the expansion of $1/p$, where $p$ is prime, is $p-1$ for sufficiently large ...