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

51
votes
5answers
5k views

Is this of any real importance to the mathematical scientific community?

I'm a 31 year old engineer, and I've recently came up with a way to exactly predict the probability of the number of prime numbers between two different integers. For example using my way, the number ...
1
vote
2answers
55 views

Polynomials mod prime $p$

The problem is $5m^2+m+4 \equiv 0\pmod 7$. I am supposed to first convert it to a quadratic whose first coefficient is $1$. But the polynomial cannot be factored, so I am unsure as to how to do ...
2
votes
2answers
69 views

Determine if $n$ is prime?

If $n < 10^6$ and no integer between $1$ and $10^4$ divides $n$. Is n prime? Here is my attempt: Assume $n$ is prime. Then using trial division, $n$ must be divisible by an integer between $1$ ...
0
votes
1answer
74 views

Show that every prime factor of $4t^2 + 1$ is equivalent to 1 modulo 4

Show every prime factor of $4t^2 + 1$ is equivalent to 1 modulo 4 My working so far: I want to use the first Nebensatz, so given q is a prime factor I want to show $(-1/q)=(-1)^{(q-1)/2}=1$ as this ...
2
votes
0answers
17 views

Solution to a set of number theoretic constraints

I'm trying to prove a graph theoretic lemma for my research; I need to construct graph homomorphisms between some delicately defined graphs. I believe I can do this if (and maybe only if) I can find ...
4
votes
1answer
46 views

Quadratic bound for prime numbers

I once found the following problem meant to be solved at high-school level (some olympiad-level exercise, I guess), and I have never been able to prove it using elementary methods. Does anybody know a ...
1
vote
1answer
49 views

Is $p\in\big\{x,…,2x\big\}$ lower-bounding $p\in\big\{x^2,…,(x+1)^2\big\}$?

Is it overreaching or erroneous to consider that possibility? (Alas, I'm not a mathematician, and don't have rigorous language to talk about this.) What I want to say is: Given any even span of ...
0
votes
1answer
25 views

Convergence behavior of $\sum_p \frac{1}{p \log p}$ and generalization.

The harmonic series $$\sum_{n\in\mathbb N} \frac{1}{n}$$ is well known to be divergent. Using Cauchy condensation test one immediately sees that even $$\sum_{n\in\mathbb N} \frac{1}{n\log n}$$ is ...
2
votes
2answers
38 views

If p and q are prime which elements are in the subgroup? (GRE question)

I was just doing some practice problems in my abstract algebra book trying to get a warm up this morning, but I found a GRE problem in the problem set and I don't know how to solve it. I've tried to ...
5
votes
8answers
284 views

If a prime $p\mid ab$, then $p\mid a$ or $p\mid b$

If a prime number $p$ is a divisor of a product $ab$, $p$ has to be a divisor of $b$ or $a$. How can I demonstrate this theorem? I demonstrated this theorem on one way using Bezout's theorem in an ...
-4
votes
0answers
35 views

Number Theory questions [duplicate]

Let $a$ be an integer and $n$ a positive integer. Prove or provide a counter example to each of the following statements. (a) If $a$ ≡ ± 1(mod p) for all primes $p$ dividing $n$, then $a^2$ ≡ 1(mod ...
4
votes
1answer
109 views

Elementary proof for $\sqrt{p_{n+1}} \notin \mathbb{Q}(\sqrt{p_1}, \sqrt{p_2}, \ldots, \sqrt{p_n})$ where $p_i$ are different prime numbers.

Take $p_1, p_2, \ldots, p_n, p_{n+1}$ be $n+1$ prime numbers in $\mathbb{P} \subseteq \mathbb{N}$. $\sqrt{p_{n+1}} \notin \mathbb{Q}(\sqrt{p_1}, \sqrt{p_2}, \ldots, \sqrt{p_n})$ seems to be quite ...
7
votes
2answers
160 views

What percentage of numbers is divisible by the set of twin primes?

What percentage of numbers is divisible by the set of twin primes $\{3,5,7,11,13,17,19,29,31\dots\}$ as $N\rightarrow \infty?$ Clarification Taking the first twin prime and creating a set out of its ...
0
votes
1answer
20 views

$\sum_{n=2}^{\infty}[\log{(\log{(p_{n+1})})} - \log{(\log{(p_n)})}] - \frac{1}{p_{n+1}} = C$?

With $p_n$ prime, does the constant, $C$, exist and have a name? $$\sum_{n=1}^{\infty}[\log{(\log{(p_{n+1})})} - \log{(\log{(p_n)})}] - \frac{1}{p_n} = C$$ If not,how about a constant and function ...
-3
votes
2answers
40 views

Proof of infinite primes

Recently , I was going through some Euclid's Lemma and I read Euclid's Proof for infinite Prime . When I read it , I found it amazing. But then thinking of How to prove it in other ways .I came to ...
3
votes
0answers
62 views

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 ...
1
vote
1answer
21 views

Is there exist $n_p\in\mathbb{N}$ such that $p+1\equiv 0 \mod (4n_p-p)$ for prime $p(\ge 5)$?

I am looking a proof for, Existence of a positive integer $n_p$ such that $$p+1\equiv 0 \mod (4n_p-p) $$ for each prime $p\ge 5.$ But I have no idea to get an attempt to this problem in general. ...
0
votes
2answers
68 views

If $ab = cd$ then $a^2 + b^2 + c^2 + d^2$ is composite [closed]

Let $a, b, c, d$ be non-zero natural numbers such that $ab = cd$. Prove that $a^2 + b^2 + c^2 + d^2$ is composite.
3
votes
2answers
51 views

Solving the diophantine equation $p^2+n-3=6^n+n^6$

What are the pairs ($p,n$) of non-negative integers where $p$ is a prime number, such that $$p^2+n-3=6^n+n^6$$ How can I solve this diophantine equation?
1
vote
1answer
107 views

Golbach's partitions: is there always one common prime in $G(n)$ and $G(n+6)$ , $n \ge 8$ (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 ...
0
votes
1answer
21 views

Consecutive terms which are all prime numbers but are also in AP

Let $a_1,a_2,a_3,\cdots$ be in AP with a common difference which is not a multiple of $3$.The maximum number of consecutive terms which are in AP and are also prime numbers is? I thought the answer ...
4
votes
3answers
81 views

Find $55! \bmod 61$

I am asked to find the smallest positive $x$ such that $x \equiv 55! \pmod{61}$. This invokes Wilson's theorem where $(p-1)! \equiv -1 \pmod p$. This means $60! \equiv -1 \pmod{61}$. But where to ...
4
votes
2answers
120 views

Show that $-3$ is a primitive root modulo $p=2q+1$

This was a question from an exam: Let $q \ge 5$ be a prime number and assume that $p=2q+1$ is also prime. Prove that $-3$ is a primitive root in $\mathbb{Z}_p$. I guess the solution goes ...
8
votes
2answers
416 views

Defining addition of supernatural numbers?

In the comments on this question Bill Dubuque mentions the supernatural numbers. My curiosity was piqued by the statement on Wikipedia that "there is no natural way to add supernatural numbers" and ...
1
vote
0answers
39 views

Comparative prime number theory with a small tweak

Fix $a, k \in \mathbb{N}$ relatively prime. For $x \in \mathbb{R}$ recall the function $$ \pi(x; k, a) = \sum_{\substack{p \leq x \\ p \equiv a \pmod{k}} } 1 $$ where $p$ denotes the primes. ...
2
votes
1answer
21 views

Logarithm of the n'th prime.

Let $P_n$ denote the n'th prime number. How could we conclude the following from the prime number theorem? $$ \log(P_n)=\log n + \log\log n + o(1) $$ Maybe by showing that $P_n=An\log n $ for a ...
4
votes
1answer
48 views

$A$ is a sum of two postive integer squares?

if $x,y,z,w$ be postive integer,and such $x^2+y^2$ is prime number,and $A=\dfrac{w^2+z^2}{x^2+y^2}\in N^{+}$ show that $A$ is a sum of two postive integer squares? maybe ...
6
votes
6answers
4k views

what's the difference between a rational number and an irrational number?

I tried to understand the difference between rational numbers and irrational numbers. I understand what is a rational number (a number that can be expressed as the ratio of two numbers p/q). what ...
3
votes
1answer
75 views

For which primes $p$ does $px^2-2y^2=1$ have a solution?

Let $p$ be an odd prime. If $px^2-2y^2=1$ is solvable, we can get Jacobi symbol $(\frac{-2}{p})=1$, so $p=8k+1,8k+3$. But when $k=12$, $p=97$, the Pell equation $97x^2-2y^2=1$ is unsolvable. I think ...
1
vote
3answers
47 views

How can i solve this diophantine equation:$x^2-(6p-4q)x+3pq=0$?

I found this diophantine equation $$x^2-(6p-4q)x+3pq=0$$ (p and q both prime numbers) and i posted my answer but i want to know if there are other methods to find the solutions of this equation. What ...
13
votes
3answers
797 views

How to prove Chebyshev's result: $\sum_{p\leq n} \frac{\log p}{p} \sim\log n $ as $n\to\infty$?

I saw reference to this result of Chebyshev's: $$\sum_{p\leq n} \frac{\log p}{p} \sim \log n \text{ as }n \to \infty,$$ and its relation to the Prime Number Theorem. I'm looking into an ...
2
votes
1answer
71 views

How does $\sum p(k)$ grow asymptotically where $p(k)$ is the smallest prime factor of $k$?

Define $p(k)$ to be the smallest prime $p$ dividing $k$. Define $A(n)=\sum_{k=2}^n p(k)$. How does $A(n)$ grow asymptotically? I am wondering how exactly the naive algorithm for finding all primes ...
-1
votes
0answers
595 views

$ (x+y) \geq (p_n +2) $?

I recently worked on a previous idea of mine (a prime number inequality, which I had posted in this community but didn't know Latex then and couldn't discuss it's proof). I was wondering how powerful ...
-2
votes
3answers
303 views

Goldbach's conjecture is wrong?! [closed]

I apologize for this very unprofessional post, but I have a lot of obligations and just I did not found the time to nicely format "my theory".I've been thinking about Goldbach hypothesis and maybe I ...
17
votes
2answers
310 views

An interesting table of Prime Generating polynomials similar to $n^2+n+41$?

Here is some data on quadratic prime generating polynomials of a particular form. Kindly look at the questions given below it. $$\begin{array}{cccc} \text{#} & P(n)=an^2+bn+c\,; & d = ...
2
votes
1answer
82 views

A generalization of Goldbach's conjecture?

In a previous question I asked about a counterexample for an observation I did about the Goldbach's comet: it seems that there is always common prime shared between the Goldbach's prime pairs of the ...
1
vote
0answers
38 views

Is there a standard notation for the sequence of sorted exponents in the prime power factorization of a number?

Given some $n \in \mathbb{N}$, is there a name or notation for any/all of the following? The set of all factors $F(n)$ of $n$ (including 1 and $n$). The ascending sequence of non-unique prime ...
25
votes
6answers
3k views

Percentage of primes among the natural numbers

How high is the percentage of primes in $\mathbb{N}$? ($\mathbb{N} := \lbrace { 1, 2, 3, \ldots \rbrace }$ ; a prime is only divisible by itself and 1 in $\mathbb{N}$) The percentage has to be lower ...
11
votes
1answer
469 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, ...
2
votes
2answers
97 views

Growth rate of product of smallest prime factors

For $n\in \mathbb{N}$, let $p(n)$ denote the smallest prime dividing $n$. Then consider the function $f:\mathbb{N}\rightarrow \mathbb{N}$ defined by $f(n)= \prod_{k=1}^{n}p(k)$. Question: What is ...
2
votes
2answers
56 views

Coprime numbers - Need help with proof

Let $a \in \mathbb{Z}$ be an odd number. Prove that the numbers $$a^{2^n} + 2^{2^n}, a^{2^m} + 2^{2^m}$$ are relatively prime (coprime) for all $m.n\in\mathbb{Z}^+$ $(m\neq n)$. Any tipps?
4
votes
1answer
58 views

How many values of $k$ satisfy $\left (\frac{k}{p}\right )=\left (\frac{k+1}{p}\right)=1$ where p is a odd prime?

The values of $k$ must be between $1$ and $p-1$ this means : $$k\in\left\{1,2,\cdots,p-1\right\}$$ The question: Given an odd prime $p$ What is the number of elements ...
2
votes
3answers
80 views

Similar to Goldbach Conjecture

I have thought of a conjecture similar to Goldbach Conjecture. I have shown the result to be true with a program in C++ up until $n=30000$. $\forall n>2$ with $n$ even,there exists two primes ...
2
votes
1answer
90 views

$\lim_{n\to\infty}\left(\frac{\log(p_{n+1})}{\log(p_n)}\right)^n = C?$

There is a conjecture (which is weaker) related conjecture to Firoozbakht's conjecture (see OEIS A182514 Commments) which states (and define $L_n$): $$L_n := ...
2
votes
1answer
27 views

A question about step in the proof of Selberg's formula

Recently I've found the following paper, discussing and proving Selberg's symmetry formula: http://www.math.uchicago.edu/~may/VIGRE/VIGRE2006/PAPERS/Balady.pdf My question concerns proofs of ...
44
votes
12answers
6k views

Proof that every number ≥ $8$ can be represented by a sum of fives and threes.

Can you check if my proof is right? Theorem. $\forall x\geq8, x$ can be represented by $5a + 3b$ where $a,b \in \mathbb{N}$. Base case(s): $x=8 = 3\cdot1 + 5\cdot1 \quad \checkmark\\ x=9 = 3\cdot3 ...
0
votes
1answer
37 views

change the order of the digits of a prime number

What is prime numbers called, that if you arbitrary change the order of its digits, you will only get another prime number. For example 79 (79 is prime number as well 97) or 199 (199, 919, 991 is ...
2
votes
2answers
52 views

If $n$ is a positive integer such that $2^n+n^2$ is a prime number , then is it true that $6|n-3$ ?

If $n$ is a positive integer such that $2^n+n^2$ is a prime number , then is it true that $6|n-3$ ? Trivially $n$ cannot be even , so this leaves us only with the possibilities $n \equiv1,3,5( \mod 6) ...
1
vote
3answers
124 views

Solve $x^p + y^p = p^z$ when $p$ is prime

Find the solutions in positive integers of $x^p + y^p = p^z$, where $p$ is a prime number. Particular case $p=2$: For $z=0$ there are no solutions. For $z=1$ the only solution is $x=y=1$. For ...
0
votes
1answer
18 views

Relation between LCM of terms of sequence with sum of sequence

Is there any relation between LCM of some arbitrary sequence and sum of elements of sequence ? How to find the LCM if only sequence sum is given in short time ?