Prime Definition and 756 Threads

Homework Statement Let a be an integer greater than 1. Then there exists a prime that divides a. Homework Equations A prime is an integer that is greater than 1 but not composite The Attempt at a Solution Via Contradiction S={x:x is an integer, x>1, x is not divisible by any prime...

Homework Statement If p is a prime greater than 3, then p leaves a remainder of 1 or 5 when divided by 6. Homework Equations I have been given the definition of composite which is an integer a is composite if there exist integers b and c such that a=bc where both b>1, c>1 3. My...

Claim: n! + 1 and (n+1)! + 1 are relatively prime. How can we prove this? Can we use mathematical induction? Base case: (n=1) gcd(2,3)=1 Therefore, the statement is true for n=1. Assuming the statement is true for n=k: gcd(k! + 1,(k+1)! + 1)=1 (induction hypothesis), we need to show...

PI(N) = N /{A * LOG(N)^2 +B * LOG(N) + C}. Note: LOG(N) is the common log. This formula works for N up to 10^23. The accuracy depends on the number of digits after the decimal point in the coefficients A, B & C. I used a Lotus123 spreadsheet to calculate them. My calculated values are...

Let p be a prime number and x be some positive integer less than p. How do I prove that there exist integers a and b such that 1 = ax + bp

Let p be a prime number. Let A be an integer divisible by p but B be an integer not be divisible by p. Let A/B be an integer. How do I prove that A/B is divisible by p? This sounds like a simple question but I just can't get it. I'm doing it in relation to proving Fermat's little...

'Prove that if a finite group G has only one maximal subgroup M, then |G| is the power of a prime' I've somehow deduced that no finite group has only one maximal subgroup, and I'm having trouble seeing where I went wrong. This is what I have: Let H_1 be a subgroup of G. Either H_1 is...

Determine all possible even perfect number(s) C such that each of C-1 and C+1 is a prime number.

Is there a name for product of the unique prime factors of a number? For example, for the number 12, it would be 3x2=6. Thanks.

Hello, I am looking into proving that the Chinese Remainder Theorem will work for two pairs of congruences IFF a congruent to b modulo(gcd(n,m)) for x congruent to a mod(n) and x congruent to b mod(m). I have gotten one direction, that given a solution to the congruences mod(m*n), then a...

First, remember that the OP is new to pure number theory. We all know that not every number that follows the formula 2n-1 is prime. My question is, without using trial and error, how would you prove or disprove this statement? "All numbers that obey the formula 2n-1, when n is a real...

Hi all, I've been staring at this question on and off for about a month: Suppose that p is an odd prime, and g and h are primitive roots modulo p. If a is an integer, then there are positive integers s and t such that a \equiv g^s \equiv h^t mod p. Show that s \equiv t mod 2. I feel as...

Homework Statement 2.4 Show that x > 1 is prime, iff x doesn't have any divisor t; where 1 < t \leq \sqrt{x}. It is given that x,t \in N. Homework Equations ? The Attempt at a Solution The "iff" thing makes me think; what can I do to show this? I have to show that x (x can be 2, 3...

Homework Statement 1- write an algorithm to find out whether an input is a prime number or not. 2-write aa program to compute square root by Newton method 3-write an algorithm to show an input is perfect or not. 4- Homework Equations The Attempt at a Solution

Homework Statement Use the Fermat test to show that 513 is not a prime number. Homework Equations The Attempt at a Solution What i have so far is: n=513 Then i pick an 'a' with 1<a<n Let a=8 So i need to compute a^(n-1) mod n -> 8^512 mod 513 If 8^512 is not...

Homework Statement Let's take a prime number p not equal to 5. Now let's take three integers a,b,c. Prove that if p | (a + b + c) \wedge p | (a^5 + b^5 + c^5), then p | (a^2 + b^2 + c^2) \vee p | (a^3 + b^3 + c^3) Homework Equations I think: (a + b + c)^2 = a^2 + b^2 + c^2 +...

I was trying to prove the statement "If (2^n)-1 is prime then n is prime". I've already seen the proof using factorisation of the difference of integers and getting a contradiction, but I was trying to use groups instead. I was wondering if it's possible, since I keep getting stuck. So far I've...

Homework Statement You bring the drinks for your soccer team every sixth game. Every third game is a home game. How many times will you bring the drinks to a home game if you have a 20 game season? Homework Equations i did the problem in my head, but I need to show my work to get the...

Homework Statement In one part of a musical composition, the triangle player in an orchestra plays once every 12 beats. The tympani player plays once every 42 beats. How often do they play together? Homework Equations don't have any The Attempt at a Solution Insufficient...

Homework Statement Presidential elections are held every four years. Senators are elected every 6 years. If a senator was elected in the presidential election year of 2000, in what year would he or she campaign again during a presidential election year? Homework Equations dont know...

Homework Statement Margo has piano lessons every two weeks. Her brother Roberto has a soccer tournament every three weeks. Her sister Randa has an orthodontist appointment every four weeks. If they all have activities this Friday, how long will it be before all of their activities fall on...

Hi, on my site http://ilario.mazzei.googlepages.com/home I've published a pdf containing the prime numbers distribution in R Ilario Mazzei

Hi, on my site http://ilario.mazzei.googlepages.com/home I've published a pdf containing the prime numbers distribution in R - Part I Ilario Mazzei

Homework Statement Show that there are exactly two minimal prime ideals in k[X,Y]/<XY>. P is a minimal prime ideal if it is prime and every subset of P that is a prime ideal is actually P. k is a field. The Attempt at a Solution Prime ideals of k[X,Y] are <0> and <f> for irreducibles...

if p is any prime other than 2 or 5, prove that p divides infinitely many of the integers 9, 99, 999, 9999, ... If p is any prime other than 2 or 5, prove that p divides infinitely many of the integers 1, 11, 111, 1111, ... Is there a way to do this problem using modular arithmetic? Thanks

Hi all, I'm reviewing some problems that involve finding the prime factors of composite numbers (e.g. prime factors of 44 are 2, 2, 11) and I had two questions about prime numbers and factors of composite numbers: 1) Is there some sort of quick test to tell if a number is prime? Or, does...

Homework Statement Show that Z/mZ X Z/nZ isomorphic to Z/mnZ iff m and n are relatively prime. (Using first isomorphism theorem) Homework Equations The Attempt at a Solution Okay, first I want to construct a hom f : Z/mZ X Z/nZ ---> Z/mnZ I have f(1,0).m = 0(mod mn) =...

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...

Homework Statement If p>=5 is prime, prove that p^2 +2 is composite. Homework Equations If we took p and divided it by 6 we would get remainder possibilities of 0, 1, 2,3,4,5 The Attempt at a Solution p=6q p^2=36q^2 P^2=6(r) P^2+2=6r+2=2(3r+1) composite p=6q+1...

http://forums.anandtech.com/messageview.aspx?catid=50&threadid=2331345&enterthread=y" I discovered this sieve a couple of weeks ago and could not find anything similar in the liturature. I made a very ugly proof and sent it off to JAMS, where it was rejected immediately. Since then, we...

Since prime colors cannot be created by any combination of any other colors, and since composite colors are combinations of the prime colors by definition, does this mean that the prime colors could form a basis for a space of colors? If you split up a color into components, like real vectors...

Homework Statement a, b, P, and any other numbers introduced are members of the integer set. If P is known to be a prime number, and ab can be divided by P, then prove that either a or b can be divided by P. Homework Equations All properties of real numbers. Need not be explicitly...

Is there a longest repeated sequence (congruency) in the prime counting function \pi (x) (that which gives the number of primes less than or equal to x)? Recall that \pi (x) , although infinite, may not be random, and itself starts out with an unrepeated sequence \pi (2)=1 and \pi (3)=2...

Can it be proven that the number of prime pairs with a difference of two (that is, primes separated by only one even number) approaches infinity?

are prime fractals, or have a fractal geometry ?? my idea is, if we consider the geometry of primes could we conclude they form a fractal ? , for example if we represent all the primes using a computer, it will give us a fractal pattern. according to a paper...

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 =...

Is there a function f(x) that will give the average number of prime factors for x_1 0<x_1<x, in a way similar to the way that Li(x)/x gives the approximate odds that a number from 0 to x is prime?

On average, at least how many factors must one try dividing a number N by to decompose it into primes?

Does either \frac{\prod_{2N=n}^\infty{p_n}}{\prod_{2N-1=n}^\infty{p_n}} or \frac{\sum_{2N=n}^\infty{p_n}}{\sum_{2N-1=n}^\infty{p_n}} converge, diverge or oscillate, where N are the natural numbers, and pn is the nth prime?

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.

Somewhere between brute force and Mersenne derivation of primes is the formula I found, \prod_{n=1}^Np_n-1=p_Z I guess it would generate more primes pZ than Mersenne in a given interval, but requires knowledge of all primes to pN, the Nth prime. It may produce only primes, rather than...

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...

\binom{n+1}{k+1}=\binom{n}{k}+\binom{n}{k+1} I'm not sure how to prove this. However...does this work: If p is a positive prime number and 0<k<p, prove p divides \binom{p}{k} Can't I just say that if that binomial is prime, this means that it is only divisible by p and 1 (since...

my question is, let us suppose we can find an expansion for the prime number (either exact or approximate) \pi (x) = \sum _{n=0}^{\infty}a_n log(x) and we have the expression for the logarithmic integral Li (x) = \sum _{n=0}^{\infty}b_n log(x) where the numbers a(n) and b(n)...

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?

i am sorry guys, the last time i posted this problem it was completely different but this time if we Let x12+x22=1 be a unit circle upon a finite field Zp where p is prime. Is there any algorithm which can give all the possible solutions (x1,x2) an element of Zp*Zp as well as the total number of...

is there any other general form of prime no.s known except 6n+/-5 and 6n+/-1. is there any general form of n such that 6n+/-5 or 6n+/-1 is a composite no.?

As I understand it, the proof that there are an infinite number of prime numbers is that if you take the factorial of the largest prime you can think of, say P!, it will be divisible by every number up to P!, as P! is equal to the product of all those numbers. As all numbers are divisible by...

Homework Statement Let n be a part of the natural numbers, with n>=2. Consider the numbers [n factorial +2 ], [n factorial + 3], ..., [n factorial + n]. Prove that none of them is prime, and deduce that there are arbitrarily long finite stretches of consecutive non-prime nummbers in the...

Homework Statement Given: For all n > 2, there exists a prime p : n < p < n! (given hint: Since n>2, one has n!-1>2 and therefore n!-1 has a prime divisor p.) The Attempt at a Solution I made an attempt at doing the proof by induction, as the previous question was by induction...