Prime Definition and 756 Threads

Homework Statement "If no prime number ##p## divides a hypothetical solution ##(x,y,z)∈ℕ×ℕ×ℕ## to the equation ##x^3+y^3=z^3##, prove that exactly one of x, y and z is even." Homework Equations Given: ~##∃p:(\frac{x}{p},\frac{y}{p},\frac{z}{p})∈ℕ×ℕ×ℕ## such that ##x^3+y^3=z^3##. In other...

It's very interesting that primes are considered the atoms of numbers, but yet, 1 isn't prime. It makes sense because one isn't an "atom" for numbers in the multiplication sense. That is, 6=2*3 =1*2*3 =1*1*2*3 =1*... So clearly 1 cannot be prime. But in the addition sense, all numbers are made...

Homework Statement Prove: If p is prime and m, n are positive integers such that p divides mn, then either p divides n or p divides m. Is anyone willing to look through this proof and give me comments on the following: a) my reasoning within the strategy I chose (validity, any constraints or...

I was reading Armstrong's Groups and Symmetry the other day and saw this table. It has beautiful symmetry. It is the the prime numbers multiplied modulo 8. It creates one of the most elegant things I've ever seen. What is so special about modulo 8 that creates such a symmetric matrix of primes?

Infinity is both a number and a concept. I asked my 10 year old niece what kind of number infinity might be and she said, "It's a composite number." But I want to think about weather infinity is a prime number? Clearly if you divide infinity by any number, you get infinity. Also if you divide...

Hello! Can anyone help me with this problem? If H is a subgroup of prime index in a finite group G, show that either N(H)=G or N(H) = H. Thank you!

Let $p$ be a prime number exceeding $5$. Prove that there exists a natural number $k$ such that each digit in the decimal representation of $pk$ is $1$ : $pk = 1111...1$

Hi everyone, been a busy week and I've got midterm cal 2 on monday so i need to get this done as fast as possible so i can focus on my math. I've got two assignments each requiring me to input 819877966 as a user number and its two parts 1.) is to break the usernumber into individual numbers...

Homework Statement Let n be an integer. Prove that the integers 6n-1, 6n+1, 6n+2, 6n+3, and 6n+5 are pairwise relatively prime. Homework EquationsThe Attempt at a Solution I tried to prove that the first two integers in the list are relatively prime. (6n-1)-(6n+1)=1 (trying to eliminate...

I am reading Joseph J. Rotman's book: Advanced Modern Algebra (AMA) and I am currently focused on Section 5.1 Prime Ideals and Maximal Ideals ... I need some help with understanding the proof of Theorem 5.12 ... ...Theorem 5.12 reads as follows: In the above text Rotman writes the following:"...

I am reading Joseph J. Rotman's book: Advanced Modern Algebra (AMA) and I am currently focused on Section 5.1 Prime Ideals and Maximal Ideals ... I need some help with understanding the proof of Theorem 5.12 ... ...Theorem 5.12 reads as follows: In the above text Rotman writes the following:"...

Hello I'm hard at work trying to find a pattern for the prime numbers and this keeps cropping up. To be honest though, to me it comes across like pseudo science. I mean I never really hear people talk about it. This seems an obvious thing to look into but I don't know anyone who does. Prime...

Mod note: moved from a homework section What properties do prime numbers exhibit which can be used in proofs to define them? Like rational numbers have a unique property that they can be expressed as a quotient of a/b. Even numbers have a unique property of divisibility by 2 and thus they can be...

I was examining the AKS and discovered this conjecture. Please prove the following true or false. Let n be an odd integer >2 then n is prime IFF $\left( \begin{array}{c} n-1 \\ \frac{n-1}{2} \\ \end{array} \right) \text{ $\equiv $ } \pm 1$ mod n

I'm solving a problem, and the solution makes the following statement: "The common difference of the arithmetic sequence 106, 116, 126, ..., 996 is relatively prime to 3. Therefore, given any three consecutive terms, exactly one of them is divisible by 3." Why is this statement true? Where does...

Hello! (Wave) Could you explain to me why the prime subfield of any field $F$ could be isomorphic to $\mathbb{Q}$ ? How do we find the prime subfield?

First a definition: given a natural number ##a_na_{n-1}...a_0##, a subnumber is any number of the form ##a_k a_{k-1}...a_{l+1}a_l## for some ##0\leq l \leq k \leq n##. I think an example will be the easiest way to illustrate this definition: the subnumbers of ##1234## are...

Is there any prime number pn, such that it has a relationship with the next prime number pn+1 p_{n+1} > p_{n}^2 If not, is there any proof saying a prime like this does not exist? I have the exact same question about this relation: p_{n+1} > 2p_{n}

On page 284 Dummit and Foote in their book Abstract Algebra define a prime element in an integral domain ... as follows: My question is as follows: What is the definition of a prime element in a ring that is not an integral domain ... does D&F's definition imply that prime elements cannot exist...

On page 284 Dummit and Foote in their book Abstract Algebra define a prime element in an integral domain ... as follows:My question is as follows: What is the definition of a prime element in a ring that is not an integral domain ... does D&F's definition imply that prime elements cannot exist...

Hi All. I have a doubt concerning the limit: $$ \lim_{n \to \infty} \frac{\pi (n)}{Li(n)} = 1 $$. This mathematical statement does not imply that both functions converge to the same value. The main reason is that both tend to infinity as n tend to infinity. I would like to ask you if it is...

I already know the answer to this easy question. Just curious what people know on the internet. A 36 volt DC electric motor, model P66SR274 is rated for 1865 watts (2.5 HP) at 2000 RPM, and its output shaft is to be direct coupled to a shaft that is 25mm in diameter. The power supply is 3X, 12...

giving : (1) $p$ is a divisor of $\dfrac{2000!}{1000!1000!}$ (2) $p$ is a prime (3) $p$ is a 3-digit number find $max(p)$

Hey! :o I want to show that the ideals $(x)$ and $(x,y)$ are prime ideals of $\mathbb{Q}[x,y]$ but only the second one is a maximal ideal. We have to show that $\mathbb{Q}[x,y]/(x)$ and $\mathbb{Q}[x,y]/(x,y)$ are integral domains, right? (Wondering) How could we show it? Could you give me...

Hey! :o I want to find the prime and maximal ideals of the ring $\mathbb{Z}_{12}$. Could you give me some hints what we could do to find them? (Wondering)

Solve the equation np_n+(n+1)p_{n+1}+(n+2)p_{n+2}=p^2_{n+2} where n\in \mathbb N^* and p_n , p_{n+1} , p_{n+2} are three consecutive prime numbers. ------------------------------------- A solution is n=2,p_2=3,p_3=5,p_4=7. May be other solutions?

Hello, Say I have some integer n in some interval such that, gcd(n, k) = gcd(n + 1, k) = 1, for some composite odd integer k >= 9 I want to know if such n exists in that interval. To know that one exists suffices. I have tried to think in terms of modular arithmetic where for all primes in k...

Homework Statement Write a program that will print all highly prime numbers from the input interval <a,b>. Prime number is highly prime if deletion of every digit from right is a prime. Example: 239 is highly prime because 239,23,2 are primes. 2. The attempt at a solution Could someone point...

Is $2017^4+4^{2017}$ a prime?

Is $3^{2520}+4^{4038}$ a prime?

Hello I am currently learning some of the basics of number theory, and struggling to understand this Theorem. Could someone please explain it with maby a simple example? :) THRM:(Number of polynomial zero mod p and H) Let p be a prime number and let H be a polynomial that is irruducible modulo...

Homework Statement If gcd(f(x),g(x)) = 1 and m,n ∈ ℕ, show that gcd(f(x)^m, g(x)^n) = 1. Homework EquationsThe Attempt at a Solution So I had previously proved this for non-polynomials: gcd(a,b)=1 then gcd(a^n,b^n)=1 Proof: a = p1*p2*...*pn b = p1*p2*...*pm then a^n = p1^n*p2^n*...*pn^n...

I am trying to find a benchmark program that will find all of the numbers between 1 and 1000 and give a time that it takes

Homework Statement Let g(x) ∈ ℤ[x] have degree at least 2, and let p be a prime number such that: (i) the leading coefficient of g(x) is not divisible by p. (ii) every other coefficient of g(x) is divisible by p. (iii) the constant term of g(x) is not divisible by p^2. a) Show that if a ∈ ℤ...

find the highest 3 digit prime factor of ${2000 \choose 1000}$

Homework Statement Show that if a, b, n, m are Natural Numbers such that a and b are relatively prime, then a^n and b^n are relatively prime. Homework Equations Relatively prime means 1 = am + bn where a and b are relatively prime. gcd(a,b) = 1 We have a couple corollaries that may be...

for details refer http://www.nytimes.com/2016/01/22/science/new-biggest-prime-number-mersenne-primes.html?_r=0

Source: http://www.mersenne.org/ See also the press release.

Homework Statement Question: Let n> 1 be an integer which is not prime. Prove that there exists a prime p such that p|n and p≤ sqrt(n). Homework Equations Fundamental theorem of arithmetic: Every integer n >1 can be written uniquely (up to order) as a product of primes. The Attempt at a...

dear sir, i wish to know if i am correct. a^2+c can be a prime number provided if a is even then c is odd or vice versa, also a and c are not multiple of same number. and c is not a negative square of any number. finally prime number is unique combination of 1,2,and other powers of 2. each power...

Does anyone know a good resource for exercises on these topics?

For prime numbers, $a$, $b$, $c$, $a^2 + b^2 \ne c^2$. Prove this by contradiction. So, I get that $a^2 = c^2 - b^2 = (c - b)(c +b)$ And I get that prime numbers are the product of 2 numbers that are either greater than one, or less than the prime numbers. But I'm unsure how to go from here.

Homework Statement My Program is not showing the sum value or not returning it. A blank space is coming.Why that is so? Homework Equations Showing the attempt below in form of code. The Attempt at a Solution #include<iostream.h> #include<conio.h> Prime_Sum(int arr[30][30],int m, int n); void...

I have a simple algorithm that appears to generate many primes (or semi-primes with relatively large factors). By 'relatively large', I mean large in relation to inputs. I have tested this algorithm for small values, and of the forty (six-digit) numbers produced, 22 are prime, 16 are...

Would it be a useful result to know there is at least one prime between 16x^2+4x-1 and 16x^2+8x-5 for any odd natural number x?

I have figured out a formula that generates prime numbers along with the proof that all such generated numbers are primes. The way it works is that you have to input consecutive prime numbers staring from 2 and ending at some Pn. And no it's not primorial minus or plus 1. Is this of any value...

Show that if G is a finite group, then it contains at least one element g with |g| a prime number. (|g| is the order of g.) Hints only as this is an assignment problem.

My apologies for such an unorthodox question, move if necessary I've not been able to find much on this, aside from that there is some conjecture { who, I have no idea } that they have understood and cataloged prime numbers. If they cataloged prime numbers they certainly understood coprime...

Hello! I have the following problem: Three brothers are aged 6, 10 and 14 years old. Will they ever, in the future, have a prime number birthday the same year? Looking at all of the prime numbers between 1 and 100, it seems that they won't. So I guess this is the same thing as saying: are...

We have the set:S={1<a<n:gcd(a,n)=1,a^(n-1)=/1(modn)} Are there prime numbers n for which S=/0?After this, are there any composite numbers n for which S=0? (with =/ i mean the 'not equal' and '0' is the empty set) for the first one i know that there are no n prime numbers suh that S to be not...