Prime Definition and 777 Threads

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...

I am reading Joseph J. Rotman's book: A First Course in Abstract Algebra with Applications (Third Edition) ... I am currently studying Section 7.1 Prime Ideals and Maximal Ideals ... ... I need help with understanding an aspect of the proof of Proposition 7.5 Proposition 7.5 and its proof...

I am reading Joseph J. Rotman's book: A First Course in Abstract Algebra with Applications (Third Edition) ... I am currently focused on Section 3.8 Quotient Rings and Finite Fields ... I need help with an aspect of the proof of Theorem 3.110. Theorem 3.110 and its proof read as follows:In...

Hello all I have done a lot of searching for what I want in a calculator and have narrowed it down to two choices. HP prime HP 50g All my life I have used ti calculators but really want to use rpn so I am really left with only one manufacture choice. That being said my correct calculators...

I am stuck on some non-coherent fault tree analysis. I have a non-coherent fault tree for which the TOP event breaks down to TOP = AD' + DA' + A'E. These are (I think) some of the prime implicants of the fault tree. There is also another prime implicant ED'. I've been trying to work through it...

I am studying this in the context of group/ring theory, ideals etc. So I post it here and not in the number theory section. 6. Suppose gcd(a,b)=1 and c|ab. Prove That there exist integers r and s such that c=rs, r|a, s|b and gcd (r,s)=1. One of my attempts: From gcd(a,b)=1 there exist s',t'...

Homework Statement x2+16Homework Equations ? The Attempt at a Solution If you attempt to solve it with (x+4)(x+4), it results in x2+8x+16, which is not equivalent. I believe it may be prime. I am looking for the formula (if there is any) to explain this. Allow me to give an example: a2+2ab+b2...

I don't quite know where to start with this one: "A natural number p>1 is called irreducible if it has the property that, for any natural numbers a and b, p|ab always implies that either p|a or p|b (or both). Prove that if a natural number p>1 is irreducible, then it also has the property...

Homework Statement Assume that n > 1 is an integer such that p does not divide n for all primes ≤ n1/3. Show that n is either a prime or the product of two primes. (Hint: assume to the contrary that n contains at least three prime factors. Try to derive a contradiction.) Homework Equations...

Hi, I need your help with the next two problems: 1) If p is a prime number such that p\equiv{3}\;mod\;4, prove that \sqrt{-p} is prime in \mathbb{Z}[\sqrt[ ]{-p}] and in \mathbb{Z}[\displaystyle\frac{1+\sqrt[ ]{-p}}{2}] too. 2) 2) We have d > 1 a square-free integer. Consider the quadratic...

Homework Statement Prove that there are infinitely many prime numbers written ##ak+b##, with ##a,b,k## integers greater than 1 Homework EquationsThe Attempt at a Solution Please could you tell me if you agree with that proof ? By contradiction: Assume that there is an integer ##k## such that...

Find all positive integers $a$ for which $a^8+a+1$ is a prime number.

I was given this problem. For every positive even integer, n, the function h(n), is defined to be the product of all the even integers from 2 to n, inclusive. If p is the smallest prime factor of h(100)+1, then p is a) between 2 and 10 b) between 10 and 20 c) between 20 and 30 d) between 30...

Hi, I'm a civil engineer finishing up my Masters with a structural concentration. Ever since I finished undergrad school I started looking for a new calculator. I decided that my frustration of scientific calculator screens would come to an end. I purchased the Casio fx-CG10. Recently I was...

Homework Statement Problem: Prove that if p is a prime number larger than 3, then p^2 = 6k + 1 for some k ∈ ℤ. Correct Solution: "Note that if p is a prime number larger than 3, then p mod 6 cannot be 0, 2, or 4 as this would mean p is even, and cannot be 3 as this would mean p is a multiple...

Hello everyone. I am going to be purchasing a good graphing calculator soon. That being said, I will hopefully be going to college to get a degree in aeronautical engineering. So what do you all think is the best of those calculators? I have been leaning towards the Casio CP400 because of the...

I put this on mt web site. it shows a serial of prime numbers. Al http://www.maverickexperiments.com/PRIME/prime.html

I have encountered the below problem- Given, ##z(z-1)## has all prime < ##\sqrt{z} <n## , Prove(or disprove)- ## π(z)-w(z-1)-A= π(2z-1)- π(z) ## where A={0 ,1}, π (z) is the prime counting function, π(2z-1)- π(z) is the number of primes in between z and (2z-1), ##\omega(z-1)## is the number...

Hi, Can we treat prime numbers as an Ortho-normal basis of "Infinite" dimensions to represent every possible number. Treating numbers as vectors. Thanks.

So if you do a search for R-Simplexs you should find that. RSimplex(n,d)=Pochhammer(n,d)/d! Well so to does RSimplex(n,d)=If(n<d, Pochhammer(d+1,n-1)/n!, Pochhammer(n,d)/d!) Or something like that my maths package is down so I'm not sure quite how it works. Anyway the relationship between...

Let $x,\,y,\,z$ be nonzero integers, $x\ne z$ such that $\dfrac{x}{z}=\dfrac{x^2+y^2}{y^2+z^2}$. Prove that $x^2+y^2+z^2$ cannot be a prime.

Homework Statement If p and q are both greater than or equal to 5, prove that 24|p^2 - q^2 Homework Equations none The Attempt at a Solution 24 = 2^3 * 3. If p = q = 5, then 24|0. If p = 7, q = 5, then 24|24. Any other combination, p^2 - q^2 will be greater than 24. I want to show that p^2 -...

Homework Statement let m|d, n|d and gcd(m,n) = 1. show mn|d Homework Equations gcd(m,n) = d = mx + ny for x and y in integers The Attempt at a Solution d = mr d = ns 1 = mx + ny 1 = (d/r)x + (d/s)y I don't know, a bit lost, just moving stuff around and not making any real progress. Any tips?

Homework Statement if gcd(a,b) = 1, show that gcd(ac,b) = gcd(c,b) Homework Equations gcd(x,y) = xm + yn for integers n and m The Attempt at a Solution ax + by = 1 gcd(ac,b) = d gcd(c,b) = g ac = dr b = ds c = gm b = gn I've been setting up equations and rearranging things but can't make...

Homework Statement Let a and b be relatively prime. Show that gcd(a+b,ab) = 1 Homework Equations ax+by = 1 for some integers x and y The Attempt at a Solution I set gcd(a+b,ab) = d. I'm now trying to show that d = 1 using elementary algebra techniques. a+b = rd ab = sd ax + by = 1I'm kind...

Homework Statement For each of the following functions with “don’t care” conditions, draw two k-maps: one for the simplified SOP and the other for the simplified POS; circle the essential prime circles, and underline the essential prime minterms (maxterms) as described in lecture. Indicate all...

Homework Statement Why does (p*q+2)-(p+q) always give a prime number when p and q are prime? Is there a similar formula that would prove this Homework Equations That's what I'm looking for. It might have something to do with Eulers formula The Attempt at a Solution I tried to find online a...

Homework Statement Let f(n) denote the number of unique prime factors of some positive integer n > 1. Prove that f(n) \in \mathcal{O}\left(\dfrac{\log n}{\log \log n}\right) Homework EquationsThe Attempt at a Solution Since every prime number except 2 is prime, an upper bound on the number...

1. The problem statement, all variafbles and given/known data Problem: Let E be a splitting field of f over F. If [E:F] is prime, show that E=F(u) for some u in E (show that E is a simple extension of F) Homework Equations Things that might be useful: If E>K>F are fields, where K and F are...

I need to find the total number of Prime Implicants in a Kmap Is the one in red also a prime implicant?

Let $a,\,b,\,p,\,q$ be integers with $a>b>p>q>0$. Suppose that $ap+bq=(b+q+a-p)(b+q-a+p)$. Prove that $ab+pq$ is not prime.

I'm trying to do some GR self-instruction through a variety of video lectures and thought this would be a good place to seek clarification on the inevitable thorny issues. I've tried this before and didn't make it too far, but I'm trying to get back on the horse, so to speak, and give it...

I am reading D.G. Northcott's book: Lessons on Rings and Modules and Multiplicities. I am currently studying Chapter 2: Prime Ideals and Primary Submodules. I need help with an aspect of the proof Proposition 3 in Chapter 2 concerning the demonstration that all the maximal elements of \Omega...

I am reading D.G. Northcott's book: Lessons on Rings and Modules and Multiplicities. I am currently studying Chapter 2: Prime Ideals and Primary Submodules. I need help with an aspect of Proposition 3 in Chapter 2. Proposition 3 and its proof read as follows: In the last sentence of the...

both $p$ and $p^2+8$ are prime numbers prove :$p^3+4 $ is also prime

Homework Statement Is it true that for each ##n\geq 2## there are two primes ##p, q \neq 1## that divide every ##\binom{n}{k}## for ##1\leq k\leq n-1##?Examples: For ##n=6: \binom{6}{1}=6; \binom{6}{2}=15; \binom{6}{3}=20; \binom{6}{4}=15; \binom{6}{5}=6.## So we can have ##p=2## and...

I was fooling around with the Prime Zeta Function just recently. Prime Zeta Function, P(s), is defined as Σ1/(p^s), where p is each successive prime. When inputting various positive integer values for (s) on wolfram alpha, I noticed a pattern. Well, an approximate pattern, I should say. My...

Homework Statement Let p be a prime, k be positive integer, and m ∈ {1, 2, 3, ..., pk-1}. Without using Lucas' theorem, prove that p divides \binom{p^k}{m}. Homework Equations The definition of the binomial coefficients: \binom{a}{b} = \frac{a!}{b! (a-b)!} The Attempt at a Solution I've...

Hello! (Wave)I want to show that the algebraic set of $K^n$, $V$, is irreducible iff $I(V)$ is a prime ideal. That's what I have tried so far: We know that the algebraic set $V$ is irreducible iff $V$ cannot be written as $V=V_1 \cup V_2$, where $V_1, V_2$ are algebraic sets of $K^n$ and $V_1...

I am reading a graduate-level Abstract Algebra lemma on noetherian integral domain, I am bring it up here hoping for pointers. The original passage is in one big-fat paragraph but I broke it down here for your easy reading. Let me know if I forget to include any underlying lemmas, thank you for...

I am reading a graduate-level Abstract Algebra lemma on noetherian integral domain, I am bring it up here hoping for help. The original passage is in one big-fat paragraph but I broke it down here for your easy reading. Let me know if I forget to include any underlying lemmas, and especially...

Prove the following theorem: Theorem For a prime number p and integer i, if 0 < i < p then p!/[(p− i)! * i] * 1/p Not sure how to go about this. I wanted to do a direct proof and this is what I've got so far. let i = p-n then p!/[(p-n)!*(p-n)] but that doesn't exactly prove much.

Hello! (Smile) Notice that in $\mathbb{C}[X,Y,Z]$: $$V(Y-X^2,Z-X^3)=\{ (t,t^2,t^3)/ t \in \mathbb{C}\}$$ In addition, show that: $$I(V(Y-X^2,Z-X^3))=<Y-X^2,-X^3>$$ Finally, prove that the ideal $<Y-X^2,Z-X^3>$ is a prime ideal of $\mathbb{C}[X,Y,Z]$. Conclude that the algebraic set...

Just want to know if there are applications in the derivation of prime numbers. My instructor and the textbook that we are using seems to be obsessed with it, there is at least one problem about deriving prime numbers in each chapter. And also different versions like palindromic prime, emirp...

If we want to know whether a certain range of numbers, say between x and y, contain prime or not. Do we use only the division method by all the prime less than the square root of y. If all the number are divisible, then there are no prime in that region. Because according to the above method...

Does there always exist primes in between square of two consecutive prime i.e Pn-1 and Pn where Pn-1 and Pn are consecutive prime. That is, in other words, does all the odd places between Pn-1 and Pn, are not divided, by primes less than Pn or by all primes upto Pn-1.I can only check randomly...

In carrom game, we have black/white small disc pieces, just imagine we have a single piece of it on the board.We hit that pieces with a striker on one side of the four wall. And the pieces goes on hitting side of the wall, number of times. If I'm right, there cannot be a general formula...

What is a prime number A number is greater than 1 is called a prime number, if it has only two factors, namely 1 and the number itself. Prime numbers up to 100 are:2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97 Procedure to find out the prime...

An integer is prime if, and only if, n > 1 and for all positive integers r and s, if n = (r)(s), then r > 1 or s > 1. it should be if n = rs, then r great than or equal 1 or s greater than or equal 1 correct?