Prime Definition and 756 Threads

I'm trying to prove that for n>2, member of Z, exists some prime p s.t. n<p<n!. I have successfully proved it by saying there's no prime btw n and (n-1)!, but I want to prove it with my original thought: first prove for 3, then for n>3: p=1+∏pi (where pi is the ith prime less ≤n) is a prime...

For each integer n > 1, let p(n) denote the largest prime factor of n. Determine all triples (x; y; z) of distinct positive integers satisfying  x; y; z are in arithmetic progression,  p(xyz) <= 3. So far I have come up with 22k + 1, 22k + 1 + 22k, and 22k + 2 other than the solutions...

I have a theory that i need to prove but I am not quite sure how to mathematically prove that it is true. Theory: When you square a rational number, each of the prime factors has an even exponent. For example, 10 --> If i square 10, which is a rational number, =10^2 =(5^2 x 2^2)...

is it possible to find a 6 Successive numbers like x , x+1 , x+2 , x+3 ,x+4 ,x+5 such that one one is prime ? Thanks

I'm having trouble with the following question: Construct a polynomial $q(x) \neq 0$ with integer coefficients which has no rational roots but is such that for any prime $p$ we can solve the congruence $q(x) \equiv 0 \mod p$ in the integers. Any hints on how to even start the problem will be...

Homework Statement If a is a prime number, prove that √a is not a rational number. (You may assume the uniqueness of prime factorization.) Homework Equations Per the text: A positive integer a is said to be prime if a > 1 and whenever a is written as the product of two positive...

Homework Statement This is a question i just got in the coursera material. Euclid's proof that there are infinitely many primes uses the fact that if p1…,pn are the first n primes, then the number N=(p1...pn)+1 is prime. True or False. The answer was False I answered true and i THINK i...

I am trying to prove that 11 is a prime in \mathbb{Z}[\sqrt{-5}]. I noticed that \mathbb{Z}[\sqrt{-5}] is not a UFD so I cannot show that it is irreducible then conclude it is prime. I know that that an ideal is prime if and only if the quotient ring is a domain. I was wondering if it is...

Homework Statement Prove that if p is prime and p \equiv 1 (mod 4), then x^{2} \equiv -1 (mod p) has a solution (x). Homework Equations We already have proved (p-1)! \equiv -1 (mod p) Hint: Use the properties of Z_{p} - a field that partitions the integers into p equivalence classes...

"Determine the least possible value of the largest term in an arithmetic progression of seven distinct primes." I really have no clue what to do here. Is there a general tactic that you can use to do this, other than trial and error? Some experimenting gives you these of arithmetic...

I was just wondering what are some applications of prime numbers other than cryptography? Also i heard that there is no certain *equation or prediction of Prime numbers? For example, there is no way to explain prime numbers with an equation. What happens if one was able to find one...

hello, I am trying to solve this problem: If w is an even integer, then w^2 - 1 is not a prime number. my current working. prove by contradiction If w is a even integer then w^2 -1 is a prime number. if w = 2x then w^{2} -1 = 4x^{2} -1 I am not sure where to go from here, maybe congruence...

Given the first N prime numbers what is the largest gap between consecutive numbers that are relatively prime to all of them? Anyone know of a fast algorithm for calculating this?

Hello, this is rather vague but I had a lecture around a year ago about prime numbers and how a mathematician (Hardy or Euler?) found a proof to do with prime numbers and then this lead on to cryptography and internet security... That's all I can particularly remember but I'm wondering on...

Homework Statement Prove by Contradiction: For all Prime Numbers a, b, and c, a^2 + b^2 =/= c^2 Homework Equations Prime number is a number whose only factors are one and itself. Proof by contradiction means that you take a statement's negation as a starting point, and find a...

Hello everybody , I'm Adrian , new stupid among apes :biggrin: This might sound silly or obvious according to a viewer's point of view and knowledge on the matter but,is there any visible undeniable linear order or logical distinguishable pattern in the distribution of primes of which humanity...

I, retired physicist (working with high level radioactive waste regulation) and now amateur mathematician, have been looking at solutions for the Pell equation x*x-D*y*y=1, and I have in particular looked at the case D=n*n-3 which contains solutions with high values for x and y, such as for...

Lets take a prime number and raise it to m/n where m and n are coprime. x,y are coprime and I want to show that this is irrational. Proof: let's assume for the sake of contradiction that P^{\frac{m}{n}}=\frac{x}{y} P is prime and m,n,x,y are integers. no we take both sides to the...

"Every prime greater than 7, P, can be written as the sum of two primes, A and B, and the subtraction of a third prime, C, in the form (A+B)-C, where A is not identical to B or C, B is not identical to C, and A, B, and C are less than P." True?

"Every prime greater than 7, P, can be written as the sum of two primes, A and B, and the subtraction of a third prime, C, in the form (A+B)-C, where A is not identical to B or C, B is not identical to C, and A, B, and C are less than P." True?

Hello all. I came across this question which is "prove that (5^125-1)/(5^25-1) is composite". I am not even getting a clue as to how I can attack this problem. Any help would be greatly appreciated. Please note that we need to prove (5^125-1)/(5^25-1)=1+5^25+5^50+5^75+5^100 as composite and not...

Hi everybody. I would like to find a book about the Distribution of Prime Numbers and the Riemann's Zeta Function. I know about the "classical" books: 1) Titchmarsh's "The Theory of the Riemann Zeta-Function" 2) Ingham's "The Distribution of Prime Numbers" 3) Ivic's "The Riemann...

I know that one of Goldbach's conjectures is that every even number is the sum of 2 primes. So, I was wondering if there was a definite, largest prime number ever possible. I know that as a number gets larger, there are more numbers that can be tried to divide it (At least I think so), and I...

Homework Statement Show that the following conditions are equivalent for a finite group G: 1.G is cyclic and |G| = p^n where p is prime and n\geq 0 2.If H and K are subgroups of G, either H⊆K or K⊆H. The Attempt at a Solution 1 => 2. Let H,K be subgroups of G = <g> where o(g)...

Homework Statement Show that m and n are relatively prime if and only if no prime divides both. The Attempt at a Solution Now, if m and n are relatively prime, we have gcd(m,n) = 1. All the common divisors of m and n must divide gcd(m,n), but the only divisors of 1 are 1 and -1. Thus...

. . Let \ \ p_n \ \ = \ \ the \ \ nth \ \ prime \ \ number.Examples:p_1 \ = \ 2 p_2 \ = \ 3 p_3 \ = \ 5 p_4 \ = \ 7- - - - - - - - - - - - - - - - - - - - - - - - - - - - Let \ \ n \ \ belong \ \ to \ \ the \ \ set \ \ of \ \ positive \ \ integers. Prove (or disprove) the following:p_n \ +...

I am working on a hydraulic system for a vehicle of nominal weight. Of course, the hp required to keep it at reasonable highway speeds is reasonably in the 30-40 hp range. My issue is energy storage for acceleration (s). This of course is substantial (like 150hp, over 8-10 seconds). My...

Given: I understand that there would have to be the equivalent energy source to drive the pump. That I will look to later (my gut says the real issue), but for the purposes of this discussion, assume endless power from a source of electricity or hydraulic pressure/flow, and you have to build a...

A problem asks to find an abelian group V and a field F such that there exist two different actions, call them \cdot and \odot, of F on V such that V is an F-module. A usual way to solve this is to take any vector space over a field with a non-trivial automorphism group, and define r\odot \mu...

Homework Statement Define n = 3^{100}+2. Suppose x^2-53 \equiv 0 \mod n has no solution. Prove that n is not prime. Homework Equations / The Attempt at a Solution Well, I suppose that I'll have to prove that some identity which should be true for n prime is not satisfied in the above case...

Homework Statement I am working my way through the Theorem on Division by a Prime. "Let p be a prime number.Then for all integers x and y, if p divides xy, then p divides x or p divides y. The proof is being done by complete induction. Proof. Let x be a whole number p a prime numberk and y be...

This is a basic abstract algebra question. Q1. Is this (x1, x2) a prime ideal in C[x1, x2, x3, x4] ? Q2. What about this: (x1 x4-x2 x3, x1 x3-x22)? Q3. Is this a prime ideal (this is the twisted cubic in projective 3-space): (x1 x4-x2 x3, x1 x3-x22, x2 x4-x32)? Thanks everyone.

Hi there, working on Prime Number Theorem and the book gives an equality that I probably should know... \frac{1}{log(2x)}= \frac{1}{logx}- \frac{log2}{log^{2}x} + O(\frac{1}{log^{3}x}) and \frac{1}{log^{2}2x} = \frac{1}{log^{2}x} + O(\frac{1}{log^{3}x}) Not sure what kind of...

uhh! help with this proof """" then x is prime""". Homework Statement For a positive integer x≥2. "if x is not divisible by any positive integer n satisfying 2≤n≤√x then x is a prime number" a) show that the above statement is true . b) Is the statement still true if the condition...

I would like to ask if somebody can verify the solution I wrote up to an exercise in my book. It's kind of long, but I have no one else to check it for me :) Homework Statement If H is a maximal proper subgroup of a finite solvable group G, then [G:H] is a prime power.Homework Equations Lemma...

Alright. I know how incredibly inefficient this algorithm is, but I felt like giving this a whirl. #include <iostream> #include <cmath> using namespace std; bool prime(int x) { bool j = true; for (double i = 2;i == x;i++) { if (x/i == floor(x/i)) { j = false; } } return j; }...

If K is a prime is there a prime between k and 2k. Obviously this is a weaker version of a prime between n and 2n that was proved by Erdos and Chebyshev. Let's assume that their isn't a prime between k and 2k. This would imply that all the numbers between k and 2k would have to be...

Homework Statement The question is not really a question from a book but rather a statement that it makes : it says " Obviously the least divisor[excluding 1] of an integer a is prime if a itself is not prime." I kind of believe this statement but I'm having trouble proving the general case...

Hi i found a question in number theory, involving two equations, it goes as follows: Let p1, p2, p3 and p4 be 4 different prime numbers satisfying the equations 2p1 + 3p2 + 5p3 + 7p4 = 162 11p1 + 7p2 + 5p3 + 4p4 = 162 Find all possible values of p1p2p3p4. Not knowing what to do, i used the...

If R is a finite ring of of order p where p is prime, show that either R is isomorphic to Z/pZ or that xy=0 for all x,y in R I know that both R and Z/pZ have the same number of elements (up to equivalence) and that R isomorphic to Z/pZ implies R must be cyclic (I think) but am otherwise...

Hi, I am writing up a project based on an algorithm for factoring large numbers, I have reached seemingly simple point where I am stuck, I wonder if anyone can help me? I am trying to factor a large N such that N=pq for unknown primes p and q, I have described a method to find a value for...

Homework Statement Allow m,n to be two relatively prime integers. You must prove that Z(sub mn) ≈ Z(sub m) x Z(sub n) Homework Equations if two groups form an isomorphism they must be onto, 1-1, and preserve the operation. The Attempt at a...

Homework Statement Let P be a prime integer, prove that Aut(Z sub P) ≈ Z sub p-1 Homework Equations none The Attempt at a Solution groups must preserve the operation, be 1-1, and be onto and they can be called an isomorphism. Z sub p-1 has one less element in it so and all the...

for estimating Pn...

Homework Statement Prove that if p is a prime and G is a group of order p^a for some a in Z+, then every subgroup of index p is normal in G. Homework Equations We know the order of H is p^(a-1). H is a maximal subgroup, if that matters. The Attempt at a Solution Suppose H≤G and...

Sum of reciprocal of some base (I just chose e as example) to prime power? Ʃ \frac{1}{e^{p}} = \frac{1}{e^2}+\frac{1}{e^3}+\frac{1}{e^5}+\frac{1}{e^7}+\frac{1}{e^{11}}+\frac{1}{e^{13}}+\frac{1}{e^{17}}+... p\inP Brute force simulation gives me ~0.19279118970439518 Is there an...

Hi, I've got an equation stating p=a(r-1). If p represents prime number and r is a positive integer, and a is a constant, what can we conclude for the constant a? Like a $\in${-1, 1, -p, p}? I suspect this has something to do with modular arithmetic...:confused: Thanks.

Homework Statement The problem is taken straight from the Project Euler website: The prime factors of 13195 are 5, 7, 13 and 29. What is the largest prime factor of the number 600851475143 ? The answer is 6857 as I must have solved it before but I can't figure out how I worked it out...

Prove that 3n + 1 has an odd prime divisor for all natural numbers > 1. I tried using order but it didn't really get me anywhere. Would prefer hints rather than complete solutions. Thanks.

Ok here's the problem: Using wolfram the first 100 results are these heres a plot of a couple points As you can see it doesn't seem to be approaching exactly zero, even though its very similar to 1/x (exactly the same if you replace Pn with just n) Is there any way to prove whether this...