Prime numbers Definition and 156 Threads

Let p_i denote the i-th prime number. Prove or disprove that: 1)\quad \displaystyle S(n) : = \sum_{i = 1}^n \mu(p_i + p_{i + 1}) < 0 \quad \forall n \in \mathbb{N}_0 : = \left\{1,2,3,...\right\}; 2)\quad \displaystyle S(n) \sim C \frac {n}{\log{n}}, where C is a negative real constant. In the...

Hi all, I had a problem, pls help me. Let b_1 < b_2 < \cdots < b_{\varphi(m)} be the integers between 1 and m that are relatively prime to m (including 1), of course, \varphi(m) is the number of integers between 1 and m that are relatively prime to m, and let B =...

Homework Statement Find a number p and an integer d>2 such that p, p+d and p+2d are all prime. Homework Equations The Attempt at a Solution Any help with how to go about this would be appreciated, thanks.

Homework Statement Suppose p is some prime number, and G a group such that \# G = p^n with some n\in\{1,2,3,\ldots\}. Prove that the center Z(G) = \{g\in G\;|\; gg'=g'g\;\forall g'\in G\} contains more than a one element. Homework Equations Obviously 1\in Z(G), so the task is to find...

Hello, I can't get this small contest problem. How do you solve this kind of problem? Let p and q be prime numbers such that (p^2+q^2)/(p+q) is an integer. Prove p=q.

Ten distinct prime numbers, each less than 3000, when arranged in increasing order of magnitude describe an arithmetic sequence. What are these ten prime numbers?

I stumbled across this question:Suppose that a and b are relatively prime.Prove that ab and a+b are relatively prime.

The Fibonacci numbers seem intimately connected with geometry. Prime numbers appear to avoid geometrics, however. Can you give some counterexamples of this latter statement?

The series of prime numbers pn=2, 3, 5, 7, 11, 13, 17, 19, 23, 27..., and Fibonacci numbers Fn=0, 1, 1, 2, 3, 5, 8, 13, 21, 34..., suggest that Fn might be considered to surpass pn exactly at an irrational value ns such that 9<ns<10 and can be determined most exactly from both series as...

What is yhe usage of big primes in Cryptography?

Ok, here's a challenge for you guys. Lets figure out a pattern for prime numbers.

Problem 1 Homework Statement Prove that p^2 - 1, where p is a prime greater than 3, is evenly divisible by 24.Homework Equations The Attempt at a Solution p^2 - 1 can be written as (p+1)(p-1) Since p is a prime, (p+1) and (p-1) must both be even numbers. Since every third integer is divisible...

i need to calculate the sum of the prime numbers below 2 million i am completely stumped help! thanks

I was playing around trying to find a function which matches the prime numbers (futile, I know) when I stumbled upon this little thing: f(n) = n^2 - (n-1)^2 (for n>1, it seems) which gives f(2) = 3 f(3) = 5 f(4) = 7 f(5) = 9 (off by -2) f(6) = 11 f(7) = 13 f(8) = 15 (-2) f(9) = 17 f(10) = 19...

Can anyone tell me, what is so important about prime numbers in science?

reciprocals of primes summed to 1 As the other thread about sum of primes started it reminded me about the idea I had long ago. My starting point was Erathostenes sieve. It occurred to me that multiples of 2 make half of all natural numbers, multiples of 3 make 1/3 of all natural numbers and...

Homework Statement Let the prime numbers, in order of magnitude, be p1, p2 ... Prove that pn+1 ≤ p1p2...pn + 1Homework Equations The Attempt at a Solution I have no idea how to start. I think it involves reductio ad absurdum.

Hi all, Do you people know about any research concerning the number that lies around twin prime numbers? I mean: How much numbers are semi-primes, for instance. I made myself clear? Sorry for the bad grammar.

Some time ago I began playing around with packing circles and I have some questions that I am hoping someone here can help with. I have linked to three PDF files that should help in understanding my synopsis below. (You will need to click on the blank sheet and then open the PDF’s as I am...

Homework Statement If F is a finite field show that there is a prime p s.t. (p times)a+a+...+a=0 for all a in the field Homework Equations The Attempt at a Solution Well I managed to prove that there must be an a in F s.t. (prime number, call p, times)a+a+...+a=0 but I can't seem...

Find all prime numbers (p,q,r), that numbers pq+pr+rq and p^3+q^3+r^3-2pqr are divided by p+q+r

My quantum professor, as an aside challenge, asked us if we could write a program in Matlab to factorize a 32 digit number into its prime number constituents. Can anyone direct me in the right direction to research how to do this? thanks, Greg

Hello, This is my first post. Anyways, from the beginning, since I started learning the subjects at higher level, I have faced this problem - How to determine if the nos. is a prime no. ? The numbers under 100 are known to me, but if a bigger digit comes, are there any tricks to...

Homework Statement I was able to prove both of these statements after getting some help from another website, but I am trying to find another way to prove them. Can you guys check my work and help me find another way to prove these, if possible? Thanks. Part A: Show that if 2^n - 1 is...

How would I write a program that finds all the prime numbers that are less than or equal to a "user-supplied" integer N, implementing the fact that I should only be dividing N by all prime numbers less than sqrt(N)?

How do I find out if 4^2007 + 2007^4 is a prime number or not?

I have to prove that there excist an infinite number of prime numbers In that proof I apply that: n=p!+1 (where p is a prime number) this number (n) is not divisible with any prime number less than or equal to p. Why is that? Is there anyone who could please explain this to me or maybe...

1. Suppose that a and b are positive integers. Show that the following are equivalent: 1) a and b are relatively prime 2) a+b and b are relatively prime 3) a and a+b are relatively prime. 2. I know that for a and b to be relatively prime, (a,b) = 1 (that is, their greatest common divisor...

[SOLVED] A different approach to prime numbers i read something about choosing a finite set of numbers as primes and deriving the other numbers from aforementioned set so that every number is obtained by multiplying primes (the numbers you choose to be prime in your system) in every possible...

I've written an algorithm that has the following goal: finding all prime numbers up to a specified integer. I've made two different algorithms actually: on one hand, I've used the concept beyond the ancient sieve of eratosthenes; on the other, I've used a function called isprime() that tests if...

I don't know if this is the right place to put this but it seemed close enough. Anyway I wanted to know if there was anyway to check whether a number is prime or not without doing a lot of division.

I am curious as to whether this pattern will always hold true: Let's say we take the prime numbers: 2,3,5,7,11,13,17,19,23...primes and we take the square(individually) minus 1 3,8,24,48,120,168,288,360,528...p^2 - 1 Then starting with the third p^2 - 1 (24), all of the p^2 - 1 can be...

Hi, I just wondered what patterns are known about the prime numbers ? cheers roger

Anyone read this? What do you think? http://www.seedmagazine.com/news/2006/03/prime_numbers_get_hitched.php?utm_source=seedmag-main=rss

A formula of prime numbers for interval (q; (q+1)^2), where q is prime number. Let: Q_k – the multitude of first k prime numbers to some extent: Q_k = (q_0 = 1^0, q_1 = 2^n1, q_2 = 3^n2, q_3 = 5^n3, q_4 = 7^n4, … q_k = u^nk) (here the expression «_i» signifies lower index, and «^ni»...

Investigating the Diophantine equation q = \frac{n^2+1}{p}} where {p} is a prime number, n,q are integers per definition The prime numbers can be sorted into two groups Group 1 has no solution and Group 2 has the solution n = \{ a\times p - b{ \ },{ \ } a\times p + b \} {\ \ \...

Why is it so difficult to predict prime numbers? And has Riemann's conjecture been solved yet?

So I just heard about the new prime number that was discovered and for some reason got kind of interested in it. So I'm looking at prime number tables on various webpages that show the prime numbers, dates discovered, etc. I'm confused on what the "digits" column in these tables means...

I've been going over this on paper for a while now, so I was wondering if maybe you guys would be able to point me in the right direction. We're supposed to prove that if every even natural number greater than 2 is the sum of two primes, than every odd number greater than 5 is the sum of 3...

Are prime numbers more than a curiosity? I know they can be useful in encrypting data but do they have a more fundamental role in the physical world? For example,a prime number won't split into two equal integer numbers. Integers occur in quantum mechanics - our most fundamental description of...

The RSA encryption algorithm that makes use of public keys, is widely used in secure communications such as e-commerce. It depends on the fact that you can multiply two very big prime numbers to get a product, but someone else cannot get back those prime numbers (factorize the product) directly...

is it true that for any set S:={1,...,n} 2*p>n , where p is the largest prime in S? Thanks in advance :biggrin:

We didn't talk about prime numbers in my class, but several of the homework problems mention them. For instance: Prove that if every even natural number greater than 2 is the sum of two primes, then every odd natural number greater than 5 is the sume of three primes. Assume that n is an...

im making an algorithm in which i need to generate two 1024 bit prime numbers, is there a way of doing this? i read that there are some ways to see how probable it is that a number is prime, can you guys point me in the right direction?

is it true that this function: f(n) = 3^(n)+2 will give a prime number for any natural value of n?

Hi, Let's talk about Prime Numbers. Still an unsolved mystery, I don't understand why it's still unsolved. Has anyone discovered why its hard to find a pattern? Or is this a silly question?

Such as sqrt 5: (2.236067977...) Start with the fractional seeds 2/1, 9/4,... New members are generated (both numerators and denominators) by the rule new member = 4 times the current plus the previous. Which generates the progrssion 2/1, 9/4, 38/17, 161/72, 682/305, 2889/1292...

Here is a cool article about a pattern to the procession of prime numbers: http://www.nature.com/nsu/030317/030317-13.html Enjoy! :smile:

Is there any good reason to define 1 as a non-prime number?

I was told that all prime numbers (except 2 and 3) could be expressed as 6n +- 1 as long as the result divided by 5 is not another integer. Is this true? Is there a proof for this (hopefully if possible not going much beyond basic calc, I am only in calc 1).