Primes Definition and 298 Threads

where could i get some info about the function \sum_{p} p^{-s}=P(s) * the functional equation relating P(s) and P(1-s) * the relation with Riemann zeta

Is the density of primes considerably greater nearer the geometric average of two consecutive square numbers? [Think of deconstructing a square of integral area n2 into composite rectangles of diverging (n-1)(n+1), (n-2)(n+2), (n-3)(n+3)... .] This reasoning may work to a lesser yet...

i saw this conjecture on the web but do not know if is true the number of primes between the expressions x^2 and (x+1)^2 for every x or at least for x bigger than 100 is equal to the Number of primes less than 2x+1 (the x are the same)

The largest known primes for some time have all been Mersenne primes. Can it be shown either that there may exist or cannot exist unknown primes less than the largest known prime?

Can someone please tell me how to go about answering a question like this? I've been racking my brain for a long time and still don't have a clue...I guess because my background in algebra/number theory really isn't that strong. "What is the greatest integer that divides p^4 - 1 for every...

i observed that every composite no. with exception of 4 and 6 can be expressed as sum of distinct prime no.s. eg: 200=103+97 100=53+47 25=13+7+5 is this true? is there any theorem stating as such?

Homework Statement Let n and k be integers with n>=2 and k>=2. Prove that (n-1)|(n^k - 1). Hence prove that if n^k - 1 is prime then n=2 and k is prime. Homework Equations The Attempt at a Solution I think you go about this question by using proof by induction. However I am...

Before I went to bed I had an idea about integers. Is there such thing as a prime number density? I just listed 1 through 50 and found that primes aren't uniformly distributed(that I noticed). Now by typical density definition the density should be the number of primes as a function of some...

What will be the numbers of positive factors of product of n distinct prime numbers? i was able to get 2^n. pls how do i prove this?

In What is Mathematics, the author gives Euclid's proof that there are infinitely many primes by assuming there aren't infinitely many, and taking all the primes, multiplying them together (P1*P2...*Pn), and then adding 1 - showing that this is larger than the largest prime, but not divisible by...

Are primes just "random"? What are the indications that primes follow some patterns and that maybe some day someone will find algorithmically simple rules for prime properties? I was thinking that maybe one merely can prove that prime numbers exist, but they represent some highly complex...

Hi all, Supppose that n > 0 and 0 < x < n are integers and x is relatively prime to n, show that there is an integer y with the property: x*y is congruent to 1 (mod n) I have attempted the following, I am not sure if I am on the right track: 1 = xy + qn which implies 1 - xy = qn n|(1-xy)...

Homework Statement Prove that any integer n >= 2 such that n divides (n-1)! + 1 is prime. Homework Equations The Attempt at a Solution I'm having trouble getting started, I have no idea how to approach this, can someone give a hint on where to begin maybe because I'm just not...

Homework Statement Here's the problem. We define the triple primes as triples of natural numbers (n,n+2,n+4) for which all three entries are prime. How many triple primes are there? (Hint:mod 3.) (By way of contrast, it is not yet known whether the twin primes-that is, pairs (n,n+2) with both...

List all of the possible sums of prime number pairs with each element taken once. For instance: 2+3=5, 2+5=7, 3+5=8, 2+7=9, 3+7=10, 5+7=12, 5+11=16, 5+13=18 . . . Can you find significance in this progression? Have you seen this sequence before?

The twin primes 5 and 7 are such that one half their sum is a perfect number. Are there any other twin primes with this property? It works for p=5. I think it should be of the form 1/2*(p+P+2). Is this true? How can I prove it? Thx

1) Find the remainder of the division of 15! with 17 2) If (n^2)+2 prime show that 3 divides n 3)If p the smallest divisor for n show that there exist integers a and b such that an+b(p-1)=1 4) For every n>1 show that n does not divide (2^n)-1 Any help?

Amazing, but it seems true: http://mersenne.org/ is currently reporting the discoveries of the 45th and 46th known presumptive Mersenne primes.

prove that a commutative noetherian ring in which all primes are maximal is artinian.

From "The Theory of Groups" by Rotman 2.5. Prove that the multiplicative group of positive rationals is generated by all rationals of the form: \frac{1}{p}, where p is prime. ... um... no it's not. Right? How can I prove this when I don't even think it is true? I mean, for example take...

Hi, I have been searching the web for this subject to see if the formula I stumbled on is out there. This site came up often, so I registered. Working with tables of the known primes < n and sum of primes < n SumP(n), I was able to determine that SumP(n) ~ Pi(n^2). See...

in how many ways we can represent 50! as the sum of two primes?

Isn't perfect randomness an unattainable ideal? So wouldn't some sort of pattern to distribution of prime numbers (i.e. other than randomness) seem to be expected? http://en.wikipedia.org/wiki/Prime_numbers" with attention to Open Questions section.

Absolutely enormous primes are known these days. It is not possible that all primes are known up to the largest ones one sees mentioned. So how far up is EVERY prime known?

Hi all, I am for some reason interested in creative or weird proofs of the fact that there are infinitely many prime numbers. I have started writing down all of the proofs that seemed sufficiently different in the following file: http://www.ocf.berkeley.edu/~ssam/primes.pdf If you know...

Reference: [PLAIN]www.mathpages.com/home367.htm[/URL] On page 2 of reference the formula is given (x+y)^p - x^p - y^p = pxy(x=y)Q(x,y) where Q(x,y) is a homogenous integer function of degree p-3. If we insert a number of different value of p into the equation, it appears that Q(x,y) =...

If one could show that Brun's constant is irrational, would that imply that there are an infinite number of primes? I think it would since Brun's constant is the sum of a bunch of fractions, and the sum of a finite number of fractions must be rational. Thus is the sum is irrational there must...

I made this simple program to list all non-primes (ignore the first row and column of the output) and list what I call "important numbers". I have attached an output if you don't want to bother running and compiling the program. #include <iostream> #include <fstream> using namespace std...

Hi all, i understand the following however i don't know how to put this on matlab. any help or hints will be very appreciated. The following algorithm enables us to identify the prime number up to a given integer N, by eliminating all non-primes in that interval. It starts from a lower end...

I was discussing something with my friends today. If you take the product of the first n primes and add 1 will this give you a prime number? For instance: 2*3 +1 = 7 ------> prime 2*3*5 +1 = 31 -------> prime 2*3*5*7 + 1 = 2311 ------> prime Can anybody find if/where this breaks...

This is the last question in Elements of Abstract Algebra by Allan Clark. When is (q) a prime ideal in Z(\rho) (the Kummer ring) where \rho = e^{2\pi i /p}, where p and q are rational primes. This seems to be a difficult question to answer in general... since considerable effort goes into...

hello guys . question here how can i prove that there exists infinitely many primes p such that p = 3 mod 4. i have a little inkling as i know that if a,b=1 mod 4 then ab = 1 mod 4. I am guessing it would be along the lines of euclids theorem?

Find all triples of primes (p,q,r), that pq+qr+rp and p^3+q^3+r^3−2pqr are divisible by p+q+r. I really don't know how to start, (of course I've been trying)

Homework Statement Let p and q be distinct primes. Suppose that H is proper subset of the integers nd H is grou under addition that contains exactly three elements of the set {p,p+q,pq, p^q, q^p}. Determine which of the following are the three elements in H: a) pq, p^q, q^p b)p+q, pq,q^p...

After studying Cesaro and Borel summation i think that sum \sum_{p} p^{k} extended over all primes is summable Cesaro C(n,k+1+\epsilon) and the series \sum_{n=0}^{\infty} M(n) and \sum_{n=0}^{\infty} \Psi (n)-n are Cesaro-summable C(n,3/2+\epsilon) for any positive epsilon...

recently i saw on a book (Apostol Analytic Number theory if i am not wrong) the prime calculating expression \sum_{p} 10^{-p}=S where the sum was extended to all the prime numbers, if i am right S=0.2003000500007.... so knowing the value of 'S' you could get the primes, hence here...

Is the set (or should I say, "sequence"?) of prime numbers a legitimate PATTERN or something else?

Hey guys I really need some help as fast as you can give it to me. Basically I want to find a selection of 7 of the following numbers, which are primes. These selections have to add up to 100 exactly, and I know that there are 35 combinations. 2 3 5 7 11 13 17 19 23 29 31 37 41...

hi y'all after lurking a lot on this forum and searching for the answer I've got something to ask, if you get that all the non trivial zeros do lie on 1/2+ib then what? I've read a book on the riemann hypothesis but i really don't get the link between the zeros of zeta(s) and the prime counting...

Erdos noticed that \sum(-1)^n\frac{n\log n}{p_n} diverges, where pn is the nth prime. I can't prove this conclusively. All I can say is that PNT implies that p_n~nlogn and thus the series "resembles" \sum(-1)^n.

Does 'Weirstrass theorem' allow the existence of an entire function so: f(z)= g(z) \prod _p(1- \frac{x}{p^{k}}) so for every prime p then f(p)=0 , and k>1 and integer?? the main question is to see if a function can have all the primes as its real roots

I am fairly certain that \frac{n}{p_n} is not monotone for any n, but I can't give a proof of it without assuming something at least as strong as the twin prime conjecture. I was wondering if anyone has some advice to prove this using known methods?

Homework Statement Prove that \sum(-1)^n\frac{n}{p_n} converges, where p_n is the nth prime. Homework Equations The sequence \frac{n}{p_n} is definitely not monotone if there exists infinitely many twin primes, since 2n-p_n<0 for sufficiently large n, so alternating series test is out. Are...

Here is the question from our book: ------ Let F_n = 2^{2^n} + 1 be the nth Fermat numbers. Use the identity a^2-b^2 = (a-b)(a+b) to show that F_n - 2 = F_0F_1\cdots F_{n-1}. Conclude that (F_n,F_m)=1 \ \forall \ n \neq m. Show that this implies the infinitude of the primes. ------- The...

http://secamlocal.ex.ac.uk/~mwatkins/zeta/NTfourier.htm" This is along the lines of what I have suspected about the primes that there is something there that is far deeper and has a real impact on both math in general and physical reality.

Question about "primes"... Hello..i've got a question that will seem "strange" or perhaps trivial...:rolleyes: :rolleyes: why are primes so important in Number theory or in maths?..there're many primality tests but my question is ..do real primes have any importance in real life?...:frown: in...

Although Andrica's conjecture is still unsolved, I'm told that it is possible to prove that \lim\sup_{n\rightarrow\infty}\sqrt{p_{n+1}}-\sqrt{p_n}=1. Does anyone know how or can point me to a source?

Sorry i don't know if this thread should be or in the "Number theory" forum, in fact if you want to calculate the series over all primes: \sum_{p} f(x) this can be very confusing as you don't know the "density" of primes my question is if we can approximate such series by the integral...

I'm looking for an algorithm or three to use in testing for prime numbers. I'm most concerned about those representable as ints or longs (that is, less than 2^63). In that range, what tests are efficient? At the moment, I'm using a combination of: * The naive division algorithm with a...

Can someone confirm/disprove the following: If X\subset\mathbb{P} is infinite, then \sum_{n\in X}\frac{1}{n} diverges or is irrational.