Prime Definition and 777 Threads

I am looking for resources which explain the prime number theorem to 18 year old students. I am not seeking a proof of the result but something which will have an impact and motivate a student to study mathematics in the future. Can anyone provide or direct me to these resources?

I am looking for resources which explain the prime number theorem to 18 year old students. I am not seeking a proof of the result but something which will have an impact and motivate a student to study mathematics in the future. Can anyone provide or direct me to these resources?

Tony Abbott, the next Prime Minister of Australia. Congratulations to him and his party. http://tvnz.co.nz/world-news/tony-abbott-time-governing-has-arrived-5575840

Why are prime numbers important in real life? What practical use are prime numbers?

Why are prime numbers important in real life? What practical use are prime numbers?

I imagine most everyone here's familiar with the proof that there's an infinite number of primes: If there were a largest prime you could take the product of all prime factors add (or take away) 1 and get another large prime (a contradiction) So what if you search for larger primes this...

1. Homework Statement . Let p be a prime number, m a natural number and G a group of order p^m. Prove that there exists an element a in G such that ord(a)=p. 3. The Attempt at a Solution . I know of the existence of Lagrange theorem, so what I thought was: I pick an arbitrary element a (I...

Is this line of thought correct? Please correct me where I´m wrong. Will this way of finding prime factors work when A is any integer? Is there a proof for this or a proof that is closely related? Is there a way to do it that requiers less iterations? It has to be a method that requires...

PS. I don't speak or write english very well. I´m doing my best. Is it still ok to post questions? a = any integer b = any prime number a * b = c Is there a proof that "c" isn't made up by any other prime factors than the prime factors that make up "a" (except for "b")?

My calculator isn't at all happy running the likely hood of finding a prime at 10,000 digits. Since there is a correlation very close to 1/2 the number of primes for each increase of 1000 digits after 1000 digits I was thinking I could just use, 1/2^(n/1000)×1151.3 = probability of finding a...

$a,b,c \in N$ $c+1=2a^2$ $c^2+1=2b^2$ c is a prime a,b,c are roots of $ 3x^3+rx^2+sx+t=0 $ please find r and t

Hi everyone, I've been bumping on this problem for a while and wondered if any of you had any clue on how to approach it. My question is whether the following equality is possible for 4 distinct prime numbers : PxPy + Pw = PwPz + Px where Px, Py, Pw, Pz are odd prime numbers, and each...

Prove or disprove for every prime P there is a K such that 10^k=1\text{mod}P. I arrived at this statement while proving something and can't find progress here is the problem which may doesn't matter but if you wan't to find the origin [here]

Hi, I have been using, for the most part, the prime notation when I want to indicate differentiation. As off recently, I have gained more insight into Leibniz's notation. This triggered the following question: how does the prime notation indicate what we are differentiating with respect to? I...

What would be the implications of assuming the existence of an imaginary number that can divide a prime number and is related to the number it is dividing? By imaginary I mean a number that is just in our imagination and not the imaginary number "i".

How do you prove/disprove the following: For any integer n higher then 1, there exists at least one prime number in interval [n+1, n^2]?

Homework Statement Prove that their are an infinite amount of primes by observing that in the series 2^2+1, 2^{2^2}+1,2^{2^3}+1,2^{2^4}+1,... every 2 numbers are relatively prime. The Attempt at a Solution I was going to take 2 of the numbers in the series and look at their difference...

I took the prime numbers from this link: http://nl.wikibooks.org/wiki/Wiskunde/Getallen/Lijst_priemgetallen I did take the first three lines I did the following with the numbers The prime 11 = 1+1 = 2 The prime 13 = 1+3 = 4 The prime 17 = 1+7 = 8 and so on This is the result for the...

Let E(x) denote the number of numbers relatively prime to x. Please help me prove that the function E(x) is multiplicative,i.e., E(xy)=E(x).E(y)

Anyone have an idea when/where there might be accessible draft of the paper, or want to share any insights beyond the press: http://www.scientificamerican.com/article.cfm?id=first-proof-that-infinite-many-prime-numbers-come-in-pairs Terry Tao has an initial comment...

http://www.nature.com/news/first-proof-that-infinitely-many-prime-numbers-come-in-pairs-1.12989

I don't know if this has already been posted. This article is about a possible proof of the twin prime conjecture. This is a breakthrough in the field of Number Theory. First proof that infinitely many prime numbers come in pairs : Nature News & Comment

Homework Statement Consider the equation as given ##\displaystyle \frac{1}{x}+\frac{1}{y}=\frac{1}{p}## where ##x, y, z, p \in I^+## and ##p## is a prime number & ##(x,y)## represents the solution pair then A)probability x<y is 1/3 B)probability that x>y is 5/6 C)probability that x≠y is 2/3...

Homework Statement Given a number N<=1018, I need to find the largest prime number less than or equal to N Homework Equations The Attempt at a Solution I can only think of a brute-force solution i.e. iterate from N in decreasing order until you get a prime number. And to check if...

1)show that for an odd natural number x, $x^2+2=3$ mod4. 2)Deduce that there exist a prime p such that $p=3$ mod4 and p|$x^2+2$ 3)Use this to prove there are infinitely many primes p such that $p=3$ mod 4 1) is easy just writing x=2m+1 2) and 3) I don't know what to do.

This is not homework. If n is a positive odd integer then n and n+2^k are relatively prime. k is a positive integer. Let's assume for contradiction that n and n+2^k have a common factor. then it should divide their difference but their difference is 2^k and since n is odd it has...

Homework Statement Let ##R## be a PID and let ##\pi\in{R}## be an irreducible element. If ##B\in{R}## and ##\pi\not{|}B##, prove ##\pi## and ##B## are relatively prime. Homework Equations ##\pi## being irreducible means for any ##a,b\in{R}## such that ##ab=\pi##, one of #a# and #b# must be a...

Homework Statement Part b and c. http://gyazo.com/821bceafd1c49adc366c63208066bd05 Homework Equations Z[x]/I is a field ⇔ I is maximal. The Attempt at a Solution b. So do I need to show Z[x]/<x,2> = { f(x) + <x,2> | f(x) in Z[x] } is a field? That would show that I is maximal...

Dummit and Foote on page 284 give the following definitions of irreducible and prime for integral domains. (I have some issues/problems with the definitions - see below)...

sometimes in statistics there is a prime ( ' ) after the variable for moment ( u sub r ) what does that mean?

The pseudoscalar mesons have J^P = 0^- They form a nonet: for S = ±1, I (isospin) = 1/2 and so there are two particles for each value of strangeness. This account for 4 particles: the ground-state Kaons. For S=0, I can be 0 or 1. I=1 gives a triplet: \pi^\pm \mbox{ and } \pi^0. For...

Homework Statement If R=\mathbb{Q}[\sqrt{2}], then Frac(R)=\mathbb{R} Homework Equations \mathbb{Q}[\sqrt{2}]=\{a+b\sqrt{2} | a,b\in{\mathbb{Q}}\} Frac(R) is the fraction field of R is basically \{\frac{a+b\sqrt{2}}{c+d\sqrt{2}} | a,b,c,d\in{\mathbb{Q}}\}. The Attempt at a Solution Back...

Here is a link to the question: If A and B are n-square matrices then prove that AB and BA have same eigen values.? - Yahoo! Answers I have posted a link there to this topic so the OP can find my response.

Given a large number N, do we have any formula to find the number of prime factors and their sum like τ(N) and σ(N) functions? CONDITION: One should not list the factors of N or is not allowed to factorize N since afterwards it would be just a matter of counting and addition

In Dummit and Foote, Section 8.3 on Unique Factorization Domains, Proposition 10 reads as follows: Proposition 10: In an integral domain a prime element is always irreducible. The proof reads as follows: =========================================================== Suppose (p) is a non-zero...

Hey guys, just looking for an explanation for the following algorithm. It is in my textbook, and there isn't really an explanation. I don't really see how the algorithm works, but I will add what I do know, and hopefully one of you can help. Thanks. //this initial declarations produces an...

I want to solve equation x(x+1)+y(y+1)=z(z+1) over primes. I found a solution x=y=2, z=3 and I have a hypothesis that this is the only solution over prime numbers, but I cannot prove it or find any other solution. Any hints, please?

1) Find all prime ideals and all maximal ideals of \mathbb{Z}_{12}. 2) Find all prime ideals and maximal ideals of \(\displaystyle \mathbb{Z}_2 \ \times \ \mathbb{Z}_4\).

Homework Statement Claim: If n is a positive integer, the prime factorization of 22n * 3n - 1 includes 11 as one of the prime factors. Homework Equations Factor Theorem: a polynomial f(x) has a factor (x-k) iff f(k)=0.The Attempt at a Solution First, we show that (x-1) is a factor of (xn-1)...

Homework Statement Show logical notation for being prime given N=(P1, P2,...Pn) +1Homework Equations The Attempt at a Solution I came up with the following, but I am not sure if it makes sense (I used trial division) (\existsx=((Pn+1)/((M>1)\wedge(M\leq√(Pn+1))))\inN) => (Pn+1 \neg=Prime...

http://news.yahoo.com/largest-prime-number-discovered-165757465.html

I noticed there was no mathematics news subforum, so this was the next best place to put this even though math related topics aren't really discussed in the chat room. As of January 25th, 2013, $2^{57885161}-1$ is the largest known Mersenne prime and is an impressive 17,425,170 digits long...

1. Homework Statement [/b] Prove that the number 2^{2^{n}} - 1 has at least n distinct prime factors. Homework Equations Seems like I'd have to use Fermat numbers and its properties to solve this question. F(n) = 2^{2^{n}} + 1 F(n) = (F(n-1) - 1)^{2} + 1 F(n) = F(0)*F(1)*...*F(n-1) +2...

Hello! This question may seem silly. I'm a first year engineering and computer science student, not a mathematics student. I have only recently become interested in prime numbers, factorization algorithms, and prime number finding algorithms. I know only extremely elementary number theory...

Hi all, I have been asked the question by a friend of mine who was working on a computer algorithm where he needed pairs of primes to uniquely identify items in a set. What I would like to know is there a way to proof that the set of prime pairs (p, p+2) is finite or infinite. I have been...

Hi, Does anyone know how the geomagnetic prime meridian is defined, particularly relative to the Greenwich meridian? Thank you.

(For the following problem I don't just want a flat out answer, but steps and Ideas on how to solve it. The problem was given by my Universities newspaper and for solving it you get free Loot and stuff)...

Im looking through old exams for a course in Cryptography and have beaten my head against the wall for a long time on one of the questions: p = 683 is a prime, p-1 = 2*11*31. What is x = 4^11112 mod p? When i did chinese remainder theorem on primes 2,11,31 i got that x = 16 mod 682, but so...

Homework Statement Let p be a prime and let n≥2 be an integer. Prove that p1/n is irrational. Homework Equations We know that for integers a>1 and b such that gcd(a,b)=1, a does not divide b^n for any n≥ 1. The Attempt at a Solution To prove irrationality, assume p^(1/n)=a/b for...

Homework Statement I am trying to prove the following: Let G be a finite group and let \{p,q\} be the set of primes dividing the order of G. Show that PQ=QP for any P Sylow p-subgroup of G and Q Sylow q-subgroup of G. Deduce that G=PQ. Homework Equations The set PQ=\{xy: x \in P \text{ and }...