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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question on prime number
Just came across the following question:
Suppose $p$ is a prime number and $p+1$ is a perfect square. Find the sum of all such prime numbers.
This is simple and there is a unique $p$, namle ...
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vote
1answer
84 views
Solving Quadratic Congruences in P Mod P
Please help me solve the following:
$$2p^2 - 42p + 221 = 0 \mod p.$$
Just messing around with the numbers I noted the following:
$p = 0 \mod p$,
therefore:
$2p^2 - 42p + 221 = p \mod p$,
...
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3answers
39 views
Problems with proof that $p|2^m-2^n$ if $p-1|m-n$
This was a homework assignment that I have already made unsuccesfully. However, no answers were given and I'm still curious. The question is as follows:
"If $p$ is an odd prime number and $m > n$ ...
12
votes
1answer
266 views
Prime spiral distribution into quadrants
Is it known that the primes on the Ulam prime spiral distribute themselves equally
in sectors around the origin? To be specific, say the quadrants?
(Each quadrant is closed on one axis and open on ...
2
votes
3answers
92 views
Prime number divisibility
The following line is in a proof I'm reading, and I don't understand the logic:
Let $\frac{a}{b}$ be an arbitrary element ($a$ and $b$ both integers). Since $p$ is a
prime, and $p$ doesn't ...
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6answers
431 views
Is there a rational number (with denominator not greater than 200) between 15/106 and 16/113?
Is there a rational number (with denominator not greater than 200) between 15/106 and 16/113?
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3answers
257 views
Do we really know the reliability of PrimeQ[n] (for $n>10^{16}$)?
The algorithm Mathematica uses for its PrimeQ function is described on MathWorld. That web page says PrimeQ uses, "the multiple ...
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0answers
131 views
Question about recursive defined functions.
This question is about counting functions.
With counting functions $F$ I mean functions from the positive integers to the positive integers that are strictly nondecreasing and can grow no faster than ...
2
votes
1answer
78 views
Distance function: $d(x,x)$ must equal zero?
Let $p$ be prime and assume $\lVert r\rVert_{p}=p^{-k}$, if $r=p^k(m/n)$, where $m$ and $n$ are relative primes of $p$. Define $$d(x,y)=\lVert x-y\rVert_{p}$$ on $\mathbb{Q}$. Show that $d(x,y)$ is a ...
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3answers
833 views
Show that product of primes, $\prod_{k=1}^{\pi(n)} p_k < 4^n$
This an interesting problem my friend has been working on for a while now (I just saw it an hour ago, but could not come up with anything substantial besides some PMI attempts).
Here's the full ...
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2answers
338 views
Highly composite number
Definition: n is said to be a highly composite number if and only if $d(n)>d(m)$ for all $m<n$, where $d(n)$ denotes number of divisors of n.
Questions:
1) Are there any theorems about highest ...
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1answer
61 views
finite field to rational fraction
Suppose I have a number $n\in\mathbb F_p$, i.e. an element of the finite field obtained by arithmetic modulo some (odd) prime $p$. I'm looking for a way to find a simple description of $n$ as a ...
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2answers
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Many kinds of Infinitely many
Here is the sequence of the primes p =1 mod 6 (and thus p =1 mod 3) such that
$(p^{2}+p+1)/ 3$ is not prime :
37 61 67 79 109 139 151 163 181 193 211 229 277 283 307 313 331 337 349 367 373 379 397 ...
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2answers
302 views
Distribution of primes?
Do primes become more or less frequent as you go further out on the number line? That is, are there more or fewer primes between $1$ and $1,000,000$ than between $1,000,000$ and $2,000,000$?
A proof ...
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vote
1answer
119 views
Set of numbers pairwise relatively prime
Given a positve integer n, we can find infinitely many positve integers $b$ such that the $n-1$ integers in the set $\{b+1,\,2b+1,\,3b+1,\,...,\,(n-1)b+1\}$ are pairwise relatively prime.
I assume ...
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1answer
27 views
Proof of Generalized Primorial Primes
Let's call the numbers of the form $k\times p\# \mp1$, the Generalized Primorial Primes.
One can find many $k$ for a fixed $p$ such that $k\times p\# \mp1$ be prime. As an example for $p = 8933$ ...
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0answers
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intuitive meaning behind Mertens' theorem
I have just been introduced the topic of distribution of primes, big O notation and aymptotic functions so please correct me if I say something that does not make sense. I am looking to get an ...
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0answers
314 views
Why does this identity equal the number of primes?
Can someone explain why this identity gives the number of primes? I don't understand it.
$D_{0,a}(n) = 1$
$D_{k,a}(n) = \displaystyle\sum_{j=1}^{k} \binom{k}{j}\sum_{m=a+1}^{\lfloor ...
6
votes
1answer
320 views
Primes of form $x^2+x\pm k$
Perhaps someone could venture an explanation (maybe with some unproven assumptions) that makes heuristic sense of relation (1),(2) and examples below? Thanks for any insights.
Let $\pi(n) = $ number ...
2
votes
1answer
37 views
Ratio of logarithmic primes
Any help is appreciated in proving/disproving the following inequality
$$
\frac{\ln{p_{n+1}}}{\ln{p_{n}}} < \frac{n+1}{n}
$$
5
votes
5answers
211 views
prime divisor of $3n+2$ proof
I have to prove that any number of the form $3n+2$ has a prime factor of the form $3m+2$. Ive started the proof
I tried saying by the division algorithm the prime factor is either the form ...
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3answers
153 views
When a prime number p divides $ab$ then we have either p divides a or p divides b.Prove that $\sqrt {p} $ is not rational for any prime number p.
When a prime number $p$ divides $ ab $ then we have either $p$ divides $a$ or $p$ divides $b$. Prove that $ \sqrt p $ is not rational for any prime number $p$.
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2answers
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A series with prime numbers and fractional parts
Considering $p_{n}$ the nth prime number, then compute the limit:
$$\lim_{n\to\infty} \left\{ \dfrac{1}{p_{1}} + \frac{1}{p_{2}}+\cdots+\frac{1}{p_{n}} \right\} - \{\log{\log n } \}$$
where $\{ x ...
5
votes
1answer
71 views
What are primes in the form of $2^n+1$ called?
What are primes in the form of $2^n+1$ called? I know that those of form $2^n-1$ are Mersenne primes, but I'm not sure about the other ones.
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2answers
142 views
Is there a list of safe prime numbers?
I am looking for a list of precomputed safe prime numbers. Where can I get such a list?
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9answers
568 views
What are the properties of a prime number?
For instance, we know that odd numbers behave like:
$$x = 2y + 1 \quad\text{where}\quad x,y\in\mathbb Z$$
For even numbers:
$$a = 2b \quad\text{where}\quad a,b\in\mathbb Z$$
But what about prime ...
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1answer
232 views
Method to solving this proof with a java app
I'm writing a program to solve this proof, but I don't know how to go about solving it.
If anyone has some insight it would be great help. Thanks
For every odd integer $n$, $3 \leq n \leq 199$, ...
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votes
1answer
97 views
3 primes conjecture
let be $ p,q,r $ prime numbers AND 'n' an integer
is then true that we can always look for p,q,r and an integer n so
$$ p^{n}+q=r $$
$ 5+2=7$
$ 2^{3}+3=11 $
$ 3^{4}+2=83 $
abnd so on
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0answers
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Are the Prime Numbers $O(f(n))$ where $f(n)$ is some polynomial?
Are the prime number, denoted $ p(n) $, $O(f(n))$, for any polynomial $f(n)$?
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1answer
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If n is an odd pseudo prime number, then $M_n = 2^n-1$ is a larger one
I came across this Theorem in "Elementary Number theorem" by David B. Burton :
"If n is an odd pseudo prime number, then $M_n = 2^n-1$ is a larger one."
I am not able to understand why this result ...
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1answer
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Testing for convergence $\sum_{j=1}^{\infty}\frac{1}{\sum_{i=1}^{j}p_i}$
How would we test for convergence the series below?
$$\sum_{j=1}^{\infty}\frac{1}{\sum_{i=1}^{j}p_i}$$
where $p_i$ is the $i$th prime number. I'd be glad to learn an elementary way. Thanks.
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votes
1answer
117 views
Is this the way to estimate the amount of lucky twins?
To estimate the amount of prime twins between $3$ and $x$ we just take $x \prod_{p}(1-2/p)$ where $p$ runs over the primes between $3$ and $\sqrt x$. Lucky numbers are similar to prime numbers. Does ...
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0answers
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Iterate over combinations ordered by sum
I have a sorted list of a large number of primes. I want to iterate over combinations of fixed size $n$ in increasing order of their sum. Naturally the standard approach for $n=4$:
$$s_0 = \sum(A, ...
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2answers
256 views
Show that $n!+1$ has a prime factor $\;>n$; show $\exists$ infinite number of primes
I don't know how to prove this and it's really bugging me. Thanks to anybody that can help!
Let $n$ be any natural number. Prove that $n! + 1$ contains a prime factor greater than $n$ and use that to ...
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2answers
26 views
Is there a pattern (or a name and expression for the pattern) of the intervals between all primes?
With the recent interest in Mersenne primes, I got thinking whether there was any mathematical expression for the pattern of intervals (or sequence composed of interval lengths) between ordinary prime ...
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1answer
46 views
Apparent patterns in ratios of consecutive primes
I was plotting the values of $\frac{P(n+1)}{P(n)+2}$, where $P(n)$ is the nth prime number. I noticed very easily that the values seem to belong very nicely to a set of "trajectories". They clearly ...
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vote
2answers
111 views
Problem over prime numbers
Which is the largest integer $n<1000$ so that $n$, $n+2$ and $n+4$ are primes?
I have tried to solve this problem but have not reached an argument worth
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3answers
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prove , if $p,q$ be two primes with the property , $q$=$p$+1 then $p$=2 and $q$=3
prove , if $p,q$ are two primes with the property , $q$=$p$+1 then $p$=2 and $q$=3
how can we prove something like that ?
my information in number theory is not big , and i have no idea about the ...
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2answers
128 views
Finding a counterexample to a Prime Factorization Conjecture
Let $\mathbb{Z}_{\geq 2}$ be the set of natural numbers starting at 2:
$$\mathbb{Z}_{\geq 2}= \{2, 3, 4, 5,\ldots\}.$$
An natural number's prime factorization is odd if the total number of primes in ...
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0answers
80 views
Is this prime formula too general?
I managed to develop a working sequence formula for primes but I think it is too general so I wanted to post it here as a question and let the community say if we could get something from it or not.
...
4
votes
1answer
103 views
Constructing arbitrary sized Miller-Rabin Primality Test Case Numbers
The Miller–Rabin (or Rabin-Miller) primality test is an algorithm that determines whether a given number is prime.
Is it possible to construct a number that will pass an arbitrary number of ...
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1answer
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Math expression for an infinite sequence of primes
At the beginning I would like to ask if there are infinite prime numbers of the form:
$$\prod_{i=1}^{n} p_i + 1$$
where $p_i$ is the $i$-th prime number; but after a google search I found that they ...
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1answer
76 views
For which prime $p$ is $x^4 \equiv -1 \pmod{p}$ solvable?
Let $p$ be a prime. I know, due to Euler's criterion, that if $x^2 \equiv -1 \pmod{p}$ is solvable, then $p \equiv 1 \pmod{4}$ simply because I inspect which $p$ that are such that $(-1)^\frac{p-1}{2} ...
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1answer
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Proving finite vs infinite representation of $p/q$ in base-$b$?
Reading up on positional notation and converting between different bases, I came across this statement:
For integers p and q with gcd(p, q) = 1, the fraction p/q has a finite representation in base b ...
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1answer
53 views
What happened to the Mertens constant in the strong prime twins conjecture ??
To estimate the amount of primes in an interval $\left(2,x\right)$ one might naively sieve by computing $ x \left(1-\dfrac{1}{2}\right)\left(1-\dfrac{1}{3}\right)...\left(1-\dfrac{1}{p_i}\right)$ ...
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Prime numbers, analysis of polylogarithms
Can any interesting results concering prime numbers be obtained using the analytic properties of the polylogarithm, similar to how analytic methods are used on the zeta function to obtain results ...
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votes
2answers
446 views
Erdős and the limiting ratio of consecutive prime numbers
The following is a piece of math lore from the late forties, which was described in an Intelligencer article entitled "The Elementary Proof of the Prime Number Theorem". It reads:
Turán, who was ...
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What can primes, except 2, 3, and 5, be congruent to $\pmod {30}$?
After some trials, I found out that a prime $p \gt 5$ is congruent to $q\pmod{30}$, where $q$ is also a prime, and $1 \le q \lt 30 \;$ (i.e. $p \equiv q\pmod{30}.$
Is there a way to write a formal ...
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1answer
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Primes without Power of 2 [duplicate]
Let $x,y,k$ be nonnegative integers, with $k$ not being a power of $2$. We also know the proof for the following statement: The number $x^k+y^k$ is not prime.
I need help on the second part:
...
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Questions about prime numbers
Let $x,y,k$ be nonnegative integers, with $k$ not being a power of $2$.
Prove that $x^k+y^k$ is not prime.
Conclude that if $2^n + 1$ is prime and $n$ is not a power of $2$, then $n$ is prime.
