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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359 views

Are closed geodesics the prime numbers of Riemannian manifolds?

I wonder to what extent one can support the analogy that primitive closed geodesics are the prime numbers of Riemannian manifolds? ("Primitive": traced once, as opposed to $m$-fold for $m \ge 2$.) In ...
5
votes
3answers
494 views

Prime Number in triangle

I had a question here, the measures of the sides of a right triangle (a single unit) can be prime numbers? If they can not, why?! But, if you can, could you help me find an example?
8
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0answers
258 views

Why are minima of $(k \bmod 4)$-Prime $\zeta$ functions $|P_x(r,t)|$ more frequent for $\frac\pi2\leq t \leq \pi$?

I got these plots when I evaluate the sum of truncated $(k \bmod 4)$-Prime $\zeta$ function, i.e. $$ P_x(r,t)=P_{x;4,1}(-ir\cos t)+P_{x;4,3}(-ir\sin t)=\sum_{x\geq p\;\bmod\;4=3} p^{-ir\cos ...
7
votes
0answers
213 views

The effect of roots of Dirichlet's $\beta$ function condenses to $\frac12\left(1+ie^{i2\pi\frac{p}4}\right)$

With the help of Raymond Manzoni and Greg Martin I was able to derive an explicit formula for the number of primes of the form $4n+3$ in terms of (sums of) sums of Riemann's $R$ functions over roots ...
3
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2answers
96 views

Extending primes

This question is more of a curiosity than anything. Start with a prime number and consider concatenating digits onto the right hand side. Sometimes you can make a prime and continue the process ...
0
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1answer
56 views

A diophantine equation related to primes.

I have $2$ prime numbers $p_1$ and $p_2$. I have to find the solution of $\large{p_1t_1+p_2t_2=1}$ where $t_1$ and $t_2$ are integers. How do I do this?
3
votes
1answer
313 views

Prove that every highly abundant or highly composite number $k$ is a prime distance from the nearest primes $\ne k \pm 1$ on either side

Prove that if $k$ is highly abundant or highly composite and $q,p$ are the nearest primes with $q+1<k<p-1$, then $k-q,p-k$ are primes. This immediately implies that all highly abundant and ...
3
votes
2answers
190 views

Efficiently identifying spam honeypots

I realise that the title is computing specific, but I think the underlying problem is general - I just don't know how to phrase it more generally (which may be part of my problem). So I am asking ...
0
votes
2answers
268 views

Prove $\log_a(b)$ is irrational given that $a, b$ are positive distinct primes.

I know this is a classical proof by contradiction exercise, and there are full solutions else where, doing a quick search I didn't find any, but I would approach this question like this: Suppose ...
0
votes
1answer
69 views

Prove that all practical numbers not of the form $2^n$ are pseudoperfect

Prove that all practical numbers not of the form $2^n$ are pseudoperfect. practical - $n$ such that every smaller integer is expressible as a sum of distinct divisors of $n$ pseudoperfect - $n$ such ...
7
votes
1answer
212 views

Prime harmonic series

We have following identity: ($p$ is a prime number) $$\left(1+\frac{1}{p}\right)\sum_{k=0}^n\frac{1}{p^{2k}}=\sum_{k=0}^{2n+1}\frac{1}{p^k}$$ Now, How to derive the following inequality from the above ...
12
votes
2answers
690 views

Diophantine equation involving prime numbers : $p^3 - q^5 = (p+q)^2$

Find all pairs of prime nummbers $p,q$ such that $p^3 - q^5 = (p+q)^2$. It's obvious that $p>q$ and $q=2$ doesn't work, then both $p,q$ are odd. Assuming $p = q + 2k$ we conclude, by the equation, ...
4
votes
3answers
171 views

How quickly can we find a prime at least as great as $n$?

This may be trivial, but I'm wondering a few things. Is there an easy way to find a prime of the form $2k+1>n$ for some $n$? EDIT How quickly can we find a prime greater than a given number $n$?
7
votes
1answer
144 views

What is the smallest real $q$ such that there is always a prime between $n^q$ and $(n+1)^q?$

In this answer, it is mentioned that for $q=3$, we are guaranteed the existence of a prime between $n^q$ and $(n+1)^q$, and that it is conjectured that this is true for $q=2$. I am wondering though, ...
4
votes
2answers
130 views

Real numbers as infinte product of primes

We can uniquely write every number in $\mathbb{Q}_+$ as $\prod_{i=1}^{N} p_i^{n_i}$ where $p_i$ is the $i$th prime number and $\{ n_i \}_{i=1}^{N}$ is some finite sequence of indices, with each $n_i$ ...
13
votes
1answer
178 views

What is the significance of the power of $3$ in the sequence of primes given by $\lfloor A^{3^n}\rfloor ?$

Mill's constant is a number such that $\lfloor A^{3^n}\rfloor$ is prime for all $n$. The existence of such an $A$ was proven in $1947$. I know little about number theory, but I am curious as to why ...
2
votes
0answers
48 views

Number of primes of type 4*n +1 in a range

I want to find number of primes which are congruent 1 (mod 4) in a range [a, b]. The range can be of order $10^9$ as a and b can be from $1$ to $10^9$. I tried segmented sieve but for a range so ...
0
votes
2answers
162 views

What is mod(a,b)?

I was reading the AKS Primality Test. AKS. I could not understand the line : $(x - a)^{n} = (x^{n} - a) \pmod{(n,x^{r}-1)}$ What is $\mod{(a,b)}$ in it ?
2
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0answers
166 views

Approximation to $\pi(x)$ conjecture.

A friend conjectured that $\left[\prod_{k=1}^{a_j <\sqrt{x}} \left(1-\frac{1}{a_k}\right)\right] x$ is usually closer to $\pi(x)$ than $\operatorname{Li}(x)$ is for some (fixed) sequence of ...
4
votes
3answers
323 views

Proving that $n\mid(nCr)$ for all $r$ ($1 \leq r \leq n-1$), only if $n$ is prime

I'm trying to prove that $n\mid(nCr)$ for all $r$ ($1 \leq r \leq n-1$) if and only if $n$ is prime. Now proving that if $n$ is prime then $n\mid(nCr)$ is pretty easy, but how would you go about ...
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4answers
73 views

Determining the general form of $10^x \bmod 210$

While solving a problem I came across solving $10^x\bmod 210$ for various values of $x$. It seems that the values repeat after an interval of 6 for $x\geq4$. Can any one explain how can solve this ...
1
vote
1answer
219 views

Divisibility of multinomial by a prime number

What is the condition for divisibility of multinomial $ \dbinom {n}{x_1, x_2, \dots, x_k} $ by a prime $p$? Update: I tried to solve using a generalisation of Lucas Theorem by representing the $n$ ...
0
votes
1answer
96 views

If $K$ and $S$ are prime numbers then how can I prove that for some $n$, there exist prime numbers of the form $K+2n+2$ and $S-2n$?

If $K$ and $S$ are prime numbers then how can I prove that for some $n$, there exist prime numbers of the form $K+2n+2$ and $S-2n$?
6
votes
1answer
273 views

Let $F_n$ be a Fibonacci number and $p$ a prime. Verify that for $p \le 61$, if $p\equiv\pm1 \pmod{5}$ then $p\mid F_{p-1}$

Define the Fibonacci entry point of $p$ to be the least integer $n$ such that $p\mid F_n$ So for example, for $p = 3$ - the Fibonacci entry point is $n = 4$ since $F_4 = 3$ and obviously $3\mid 3$. ...
14
votes
4answers
698 views

Why do repunit primes have only a prime number of consecutive $1$s?

Repunit primes are primes of the form $\frac{10^n - 1}{9} = 1111\dots11 \space (n-1 \space ones)$. Each repunit prime is denoted by $R_i$, where $i$ is the number of consecutive $1$s it has. So far, ...
8
votes
1answer
231 views

Prove that $\forall p \in \Bbb P;p \ne 5,$ $F_{p^n - \left(\frac{5}{p}\right)p^{n-1}} \equiv 0 \mod p^n$

Prove that $\forall p \in \Bbb P,n \in \Bbb Z^+;p \ne 5,$ $F_{p^n - \left(\frac{5}{p}\right)p^{n-1}} \equiv 0 \mod p^n$ and $F_{5^n} \equiv 0 \mod 5^n$, where $\left(\dfrac{5}p\right)$ is the Legendre ...
10
votes
6answers
1k views

What are some easy-to-remember prime numbers? [closed]

This is a question without much mathematical value, but since I don't immediately see an answer on Google I thought I'd ask anyway ... I'm looking for some largeish (> 10,000) easy-to-remember primes, ...
1
vote
1answer
231 views

Linear equation with prime coefficient.

Suppose we have a linear equation with two variables say $x$ and $y$ and three integer coefficient $a , b$ and $c$ (constant), where $a$ and $b$ are prime all are greater than zero. $ax+by=c$ how ...
4
votes
1answer
499 views

Let $p$ be a prime and $q$ a prime divisor of $2^{p} -1$. Use Fermat's Little Theorem to prove that $q\equiv 1 (\mod \space p)$

Question continued: Hint: Consider $ord_{q}(2)$. Similarly, prove that if $r$ is a prime factor of $2^{2^{k}}+ 1 $ then $r\equiv1 (\mod \space 2^{k+1})$ I think I have the first part, however I ...
1
vote
2answers
275 views

Twin, cousin, and sexy prime property

Why the digital root of twin primes is always $(2,4) (8,1) (5,7)$? Why the digital root of two primes with difference $4$ is always $(4,8) (1,5) (7,2)$?
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2answers
121 views

Problem of Ages (Problema das Idades)

English: Somebody help me with this challenge? It's very confusing: Today, both me and my younger brother are between $10$ and $20$ years old. Also, our ages are expressed by prime numbers and the ...
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0answers
51 views

What is the signifigance of prime numbers [duplicate]

A buddy of mine said that prime numbers are valuable and people will actually pay for new, undiscovered prime numbers. He said people build computers that specifically just "mine" for prime numbers. ...
2
votes
1answer
96 views

Between Mertens' theorems

It is well-known that $$ \sum_{p\le x}\frac{\log p}{p}=\log x+O(1) $$ and $$ \sum_{p\le x}\frac1p=\log\log x+M+o(1). $$ What is the order of $$ \sum_{p\le x}\frac{\sqrt{\log p}}{p} $$ ?
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1answer
121 views

Theorem on Sum of prime factors

Robin's theorem gives an inequality for the divisor function of a number. Is there an equivalent theorem where we have an inequality for the sum of the prime factors of that number instead of the ...
1
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1answer
97 views

Prime Numbers and Primitive Roots

Let $p_1$, $ p_2$, $p_3$ different prime numbers. Let $N = p_1p_2p_3$. Given $(p_1-1)|(N-1), (p_2-1)|(N-1)$ and $(p_3-1)|(N-1)$, prove that for every number $a \in \Bbb N$ such that $\gcd(a,N) = 1$ ...
1
vote
1answer
64 views

If $p$ is irreducible and $p \not \mid a$, then $\text{gcd}(p,a)=\pm 1$.

I will be taking a Rings and Fields course in the Fall, so I figured I would read ahead in the textbook (A First Course in Abstract Algebra, by Anderson and Feil) to prepare. Recall the following ...
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2answers
708 views

Primes between $n$ and $2n$

I know that there exists a prime between $n$ and $2n$ for all $2\leq n \in \mathbb{N}$ . Which number is the fourth number that has just one prime in its gap? First three numbers are $2$ , $3$ and $5$ ...
2
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2answers
196 views

A trivial question about the $n^{\text{th}}$ prime number

I would like to ask for a little help about a very trivial question from the number theory. If $p_{n}$ is the $n^{\text{th}}$ prime number in the ascending sequence of prime numbers, show that ...
2
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1answer
67 views

A question on primes and equal products

Is the following statement true; "For any odd prime number $p$ , any set of $p-1$ consecutive integers can not be partitioned into two subsets such that the elements of the two sets have equal ...
7
votes
1answer
457 views

Newman's “Natural proof”(Analytic) of Prime Number Theorem (1980)

I am trying to understand this short proof by newmann. I faced some problems while grasping this very proof. Please help me out. 1 . I am not clear, why in step (1)'s proof he says that from unique ...
0
votes
1answer
29 views

Finding $n$ such that $\sum^{n}_{k=0} \frac{2}{p_k} = \left ( \prod^{n}_{j=0} p_j^{-1}\right) p_x$

Let $p_n$ denote the $n$th prime. Is it possible to find $n$ such that $$\sum^{n}_{k=0} \frac{2}{p_k} = \left ( \prod^{n}_{j=0} p_j^{-1}\right) p_x$$ any other way than calculating both the ...
1
vote
1answer
91 views

Sum containing primes

Can anybody compute the value of $$\sum_p\sum_{k=2}^\infty\frac{\log(p^k)}{k}-\sum_p\sum_{k=2}^\infty\frac{\sum\limits_{p^n<k}\log(p^n)}{k(k+1)}$$ I have tried a lot but cannot think about the ...
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0answers
38 views

Reasoning about $\left\lfloor\frac{p_k\#}{p_{k+1}}\right\rfloor$

This is a follow up question to my previous question. Let $$v_i = \left\lfloor\frac{ip_k\#}{p_{k+1}}\right\rfloor + c_i$$ where: $c_i \in \left\{1,2\right\}$ so that $v_i$ is odd and $v_ip_{k+1} ...
1
vote
1answer
40 views

Trying to understand why a set of residues modulo a primorial $p_k\#$ has a range of values smaller than $2p_{k+1}$

I've been reviewing the following: $$v_i = \left\lfloor\frac{ip_k\#}{p_{k+1}}\right\rfloor + c_i$$ where: $c_i \in \left\{1,2\right\}$ so that $v_i$ is odd and $v_ip_{k+1} > ip_k\# > ...
2
votes
1answer
49 views

Modulus of a large prime, a smaller prime multiplied by a positive factor n doesn't repeat.

Given a prime p1 and a prime p2 (where p2 < p1), I made the observation that with a number n (where n increases by 1 from the value 1) the equation below results in two properties: ...
4
votes
1answer
74 views

Infinitely many maximal and nonmaximal prime gaps?

This is a simple question about first occurrence prime gaps and maximal and nonmaximal prime gaps. A gap between prime numbers is maximal if it is larger than all gaps between smaller primes. My ...
3
votes
1answer
107 views

Bertrand's postulate proof

Regarding http://michaelnielsen.org/polymath1/index.php?title=Bertrand%27s_postulate I think the last inequality should be $4^{n/3}\le(2n+1)(2n)^{\sqrt{2n}}$. But even when the RHS is decreased from ...
3
votes
1answer
66 views

Why there is always a number $b \in Z$ s.t $(\frac{b}{p})=-1$ for prime $p$?

Out teacher said that for any odd prime $p$ there exists a number $b \in \mathbb{Z}$ s.t $(\frac{b}{p})=-1$ (We are talking about legendre symbol here). Intuitively I can understand why, but a short ...
2
votes
0answers
58 views

Related almost primes

Do there exist positive integers $w, x, y, z$ and an infinite set $A$ such that for all $a \in A$, $a$ is a $w$-almost prime and $b = y \cdot a + z$ is an $x$-almost prime? With $(w,x,y,z) = ...
4
votes
2answers
109 views

Finding a function whose value at $n$ is the $n^{\text{th}}$ prime

For positive integers $a$ and $b$, evaluate: $$f\left ( a,b \right )=\frac{1}{a}\sum_{j=1}^{a}\cos\left ( \frac{2\pi jb}{a} \right )$$ Hence, find a function $g\left ( n \right )$, $n \in ...