2
votes
1answer
50 views

Does there exist a finite set of polynomials which do not have roots over any prime field?

The polynomial $x^2 + 1$ has a root in $Z_p$ if and only if $p \not\equiv 3 \mod 4$, and the polynomial $x^2 + x + 1$ has a root in $Z_p$ if and only if $p \not\equiv 2 \mod 3$. So each of the ...
-5
votes
1answer
179 views

Do Prime Numbers have a Structure or do they sprout out Randomly among positive Integers? [duplicate]

Since the Order of Sequence of the Prime Numbers has not been found, it seems that all famous Mathematicians have opted for the random appearance of Primes.
31
votes
5answers
4k views

The myth of no prime formula?

Terence Tao claims: For instance, we have an exact formula for the $n^\text{th}$ square number – it is $n^2$ – but we do not have a (useful) exact formula for the $n^\text{th}$ ...
0
votes
1answer
24 views

Proving an inequality on the sum of $\log$ of primes.

Let $S(x)=\sum_{p\leq x} \ln(p)$ where $\sum_{p\leq x}$ denotes a summation over the positive prime numbers that are $\leq x$ Prove that $\forall n \in \mathbb N, S(2n+1)-S(n+1)\leq ...
0
votes
0answers
33 views

Upper bound number of distinct prime factors

I want to prove that if $\omega (n)$ is the number of distinct prime factors of $n$ for $n>2$ we have $\omega (n) \leq \frac{\ln n}{\ln \ln n} + O(\frac{\ln n}{(\ln \ln n)^2})$. How can I do this? ...
0
votes
0answers
51 views

Relation between Galois theory and Fermat primes

I am curious about a possible relation between Galois theory and Fermat primes. There is a general solution to any polynomial equation of degree less than or equal to $4$. The only Fermat primes (of ...
0
votes
1answer
41 views

Prime number theorem lemma: prove that $\psi(x)\sim\pi(x)\log(x)$

I'm trying to follow the proof in Wikipedia that the PNT is equivalent to the assertion $\psi(x)\sim x$, by proving that $\psi(x)\sim\pi(x)\log x$, which it claims is a very simple proof. One ...
0
votes
1answer
39 views

Length in time to find the longest range of primes between 2 and a 13 million character digit?

I am trying to run a program that tells me how many prime numbers there are in a range of numbers. I run it in intervals of 10,000 to 100,000. How long would the program take to determine all the ...
1
vote
0answers
39 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 ...
0
votes
1answer
27 views

Minimal distance between coprimes

For a natural number $K$, I want to choose $n$ pairwise coprime numbers all of which are bigger than $K$ such that the distance $d$ between the smallest and the largest one is minimal. For example, ...
2
votes
1answer
32 views

p-adic numbers and GCD

Given two numbers $a,b \in \mathbb{Z}$, how do we prove that the $p$-adic number of $\gcd(a,b)$ is the same as the minimum for the $p$-adic number of $a$ and the $p$-adic number of $b$? Does this ...
3
votes
1answer
48 views

Foundational proof for Mersenne primes

I know how to prove that, if $2^n-1$ is prime and $n>1$, then $n$ is prime. But how do we prove that, if $a^n-1$ is prime and $n>1$, then $a$ must equal 2?
5
votes
0answers
57 views

To solve for $x,y,n$ in non-negative integers , $\dfrac{x!+y!}{n!}=p^n$ , $p$ a given prime

Let $p$ be a given prime , then how do we find non-negative integers $(x,y,n)$ $\space$ , such that $\dfrac{x!+y!}{n!}=p^n$ ?
3
votes
1answer
92 views

Is there a prime number ending with the natural number $n$

if $n$ not is divisible by 2 or 5? Example: given 813075843967837637675737563754361301, there is a prime 20813075843967837637675737563754361301 or given ...
3
votes
3answers
103 views

How amount of Emirp Primes depends on the base of numeral system

There was a problem on searching for primes which, if their decimal notation is reverted, yield another primes, like 37 => 73 or ...
0
votes
1answer
43 views

How many terms are required to get $D$ digits of Riemann zeta prime function?

How many terms are required to get $D$ digits of Riemann zeta prime function $\zeta_p(s) = \sum_p \frac{1}{p^s}$? Sebah & Gourdon mentions that finding $\zeta_p(2)$ to 20 digits by using $\sum_p ...
3
votes
1answer
59 views

Greatest common divisor problem involving $a^p+b^p$ [closed]

Let $\gcd(a,b)=1$ for some $a,b\ \epsilon \ \mathbb{N}$. Prove that for any odd prime p: $$\gcd\left(a+b,\frac{a^p+b^p}{a+b}\right)=1,~~~~ \text{or} ~~~p.$$
0
votes
0answers
19 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 ...
2
votes
1answer
33 views

Firoozbakht's conjecture and maximal gaps

In the Wikipedia article, it seems to me as if it's implied that it is enough to check the conjecture only for maximal gaps (numbers $n$ s.t. $\forall k<n:g_n>g_k$). I.e it holds that ...
0
votes
1answer
37 views

$\left| (4 \mathbb{N} -1) \cap \mathbb{P} \right| \ = \ \infty$ where $\mathbb{P}$ is the set of prime numbers. [duplicate]

I try to show that there are infinitely many prime numbers in the set $ \{ 4n-1 \ : \ n \in \mathbb{N} \}$. I've been told that I needed to adjust Euclid's proof a bit so that it would work for ...
2
votes
2answers
44 views

Legendre's formula

Legendre's formula counts the number of positive integers less than or equal to a number $x$ which are not divisible by any of the first $a$ primes: $$\begin{align} &\phi(x,a)=\lfloor x ...
4
votes
1answer
120 views

Relationship between Primes and Fibonacci Sequence

I recently stumbled across an unexpected relationship between the prime numbers and the Fibonacci sequence. We know a lot about Fibonacci numbers but relatively little about primes, so this connection ...
4
votes
7answers
105 views

Finite sequences of prime numbers

There is a lot of prime sequences: prime numbers in a special form. For example Mersenne primes are primes of the the form $2^n-1$, or Pythagorean prime are primes of the form $4n+1$. Even primes are ...
11
votes
7answers
756 views

Define an infinite subset of primes such that the sum of reciprocals converges

How can we define an infinite subset of primes such that the sum of reciprocals converges? $S=\{p\in \mathbb{Z}^+ : p\ \text{is prime and some condition on}\ p\}$ s.t. ...
1
vote
2answers
48 views

series for $n$-th prime number and prime counting function

"Theoretical Computer Science Cheat Sheet" gives the following: $$p_n = n \ln n + n \ln \ln n - n + n \frac{\ln \ln n}{\ln n} + \mathcal{O}\left( \frac{n}{\ln n}\right)$$ $$\pi (n) = \frac{n}{\ln n} + ...
2
votes
0answers
63 views

Are there infinite many primes p such that 2p-1 is also prime?

I did a search online and found a similar notion called Sophie Germain prime, which by definition is a prime $p$ such that $2p+1$ is also prime. Sophie Germain primes are conjectured to be of infinite ...
10
votes
3answers
2k views

Why are all non-prime numbers divisible by a prime number?

In Euclid's infinite prime numbers proof, the logic is as follows: Assume a set $S$ of all prime numbers in existence is finite (there are a finite amount of primes) Then there must be a greatest ...
4
votes
3answers
76 views

Prove that there are infinitely many primes $P_i\equiv1\pmod6$

Proving that there are infinitely many primes is fairly simple: Assume that there is a finite number of primes. Let $G$ be the set of all primes $P_1,P_2,\ldots,P_n$. Compute $K = P_1 \times P_2 ...
4
votes
1answer
79 views

What are major algebraic number theory attempts, results and progressions toward Goldbach's Conjecture?

To my understanding, most progress toward Goldbach's Conjecture has been made in analytic number theory. Progress has often based on sieve, asymptotic estimation or other analytic methods. What are ...
1
vote
1answer
27 views

Can the natural numbers be defined in terms of the non-trivial zeta zeros?

Can the natural numbers be defined in terms of the non-trivial zeta zeros? Presumably they can, since $\pi(x)=\operatorname{R}(x)-\sum_{\rho}\operatorname{R}(x^\rho),$ and $\zeta(s)=\sum ...
4
votes
2answers
47 views

Heuristic for Dirichlet's Theorem on Arithemtic Progression

If we let $\pi_{a,d}(x) = \{p \leq x: p \mbox{ prime, } p \equiv a \mod{d}\}$ then it is a well known result that if $(a,d)=1$ then $$\lim_{x \to \infty} \frac{\pi_{a,d}(x)}{\pi(x)} = ...
4
votes
0answers
120 views

Differences between large numbers with many factors has little factors

I apologise beforehand for the informality and lack of precision in this question but it is that way because it comes from only an intuition, nothing more than a heuristic argument. Say one has two ...
0
votes
1answer
36 views

How would you show $\pi(x)\log(1-\frac{1}{x}) \sim \frac{1}{\log x}$

How would you show $\pi(x)\log(1-\frac{1}{x}) \sim \frac{1}{\log x}$? Would you use $\lim_{x\to \infty}\frac{\pi(x)\log(1-\frac{1}{x})}{\frac{1}{\log x}} = 1$? and how would you show this? Can you ...
0
votes
1answer
39 views

How many distinct lists of 14 integers $L=\{v_1,\ldots ,v_{14}\}$ exist satisfying $v_i \geq v_{i+1}\geq 0$ and $\sum _{i=1}^{14}(v_i) \leq 54$

I am trying to solve the following problem: I have an ordered list of integers $L = \{v_1,\ldots ,v_{14}\}$ with fourteen elements, satisfying the following two conditions: $v_i \geq v_{i+1}\geq 0$ ...
0
votes
1answer
61 views

Count numbers with prime digit

Given a number N I need to find the count of the numbers that have atleast one prime digit (2,3,5 or 7) in it. Now N can be upto 10^18.What is the best approach to solve this problem. Example : Let ...
2
votes
1answer
32 views

Finding the lowest number (or an upper bound to the lowest number) not congruent to a set of moduli

Note: if finding x is not possible, an upper bound, where there must be at least one number less than said number which is not congruent to the set, would be helpful. The set: For my purposes, the ...
4
votes
0answers
190 views

Prove modular equivalence related to carmichael function.

Statement if $n=p_1p_2p_3\cdots p_r$ (prime numbers are distinct) and $2 \le a \le n-1$ then prove that $$a^{k\lambda} \equiv \left(1+\sum_{p_i\mid\gcd(a,n)}\left(\dfrac{n}{p_i} \right) ...
2
votes
2answers
59 views

Diophantine: $px^2+2=y^2$ where $p\in \mathbb{P}$

Solve the Diophantine Equation: $px^2+2=y^2$, where $p$ is a prime number and $x,y$ integers. I tried this for ages but didn't get anywhere, but I don't know any advanced machinery since I am only in ...
7
votes
1answer
72 views

What is $\limsup\limits_{n\rightarrow\infty}\left\{\frac{p_{m+1}}{p_m}|m\in \mathbb{N},m\geq n\right\} $?

What is $$\limsup\limits_{n\rightarrow\infty}\left\{\frac{p_{m+1}}{p_m}\middle|m\in \mathbb{N},m\geq n\right\} = ?$$ where $p_i$ is i'th prime number. We know that this limsup exists because of ...
0
votes
3answers
75 views

Number theory divisibility - simple way to prove this is prime?

Suppose that $y$ is a positive integer, and $z$ is the largest factor of $y$ such that $z<y$, then let $x=y/z$. Prove that $x$ must be a prime number. Is there a simple way to solve this? It ...
1
vote
2answers
119 views

Property of set of prime numbers

let $\{p_1,p_2,p_2,\cdots ,p_r\}$ be the set of $r$($\ge2$) pair wise distinct prime numbers i.e.., $(i\ne j \implies p_i \ne p_j)$ for all $1\le i,j\le r$ ${Statement}$ : For any such ...
0
votes
0answers
45 views

Number of prime factors of Mersenne numbers

Let $p$ be a prime and let $M_p = 2^p-1$. Is it known whether the number of prime factors of $M_p$ is unbounded above as $p \to \infty$? Also do the probabilities estimating the chance that $M_p$ is ...
1
vote
1answer
33 views

Conditions for which $(\sum p_i)(\sum \frac{1}{p_i})$ over an arbitrary $i$ for a set of primes $\{p_i\}$ is unique?

I am looking for conditions (if any are needed beyond properties of primes) for which $(\sum p_i)(\sum \frac{1}{p_i})$ over an arbitrary $i$ for a set of primes $\{p_i\}$ is unique in that there is ...
6
votes
1answer
215 views

Can Fermat's little theorem be used to list primes?

I was reading about Fermat's little Theorem, which states that if p is prime, then for any integer a, $a^p-a$ would be a multiple of p. So, I started wondeing if this could be used to determine ...
2
votes
1answer
209 views

Unusual behavior of 210 and 199 regarding prime numbers

Adding 210 to 199 over and over again, you get 8 more primes that can be arranged together into a 3x3 magic square: 1669 199 1249 619 1039 1459 829 1879 409 Is there any other pairs of numbers ...
0
votes
1answer
95 views

Computing infinite product over primes

How can I compute $$ \prod_p \left(1+\frac{k}{p}\right)\exp(-k/p) $$ where $0<k<e$ and the product is over all primes $p$? Background L. G. Sathe proved [1] that there are $$ ...
2
votes
3answers
58 views

Smallest prime factor of a Mersenne number

The Mersenne numbers $M_n$ are integers of the form $2^n-1$, where $n$ is a positive integer. In the case when $n$ is a prime, are there any results known on the smallest prime factor, $p_n$, of ...
1
vote
1answer
34 views

Given a sequence, construct a function whose integral is equal to the sum of the sequence

Let $P_n$ be the sequence of prime numbers, where $P_0=2$. Given $m\in\mathbb{N}$, how can we construct $f(x)$ such that: $\displaystyle\forall{0}\leq{i}\leq{m}:f(i)=P_i$ ...
1
vote
1answer
45 views

A different way to solve Chinese remainder theorem

I'm doing my homework about Chinese remainder theorem $x = a_1(\mod n_1)$ $x = a_2(\mod n_2)$ As I know, x can be found by using: $$x=\{\sum_{i=1}^na_iN_i(N_i^{-1}(\mod n_i)) \}(\mod N)$$ with ...
1
vote
4answers
151 views

The fastest way to count prime number that smaller or equal N

I want to count all prime numbers that existing in N but I don't know how to count. Can any one tell me how to count prime numbers that are smaller than or equal to N in mathematics formal?