Prime Definition and 756 Threads

Homework Statement The problem (and its solution) are attached in TheProblemAndSolution.jpg. Specifically, I am referring to problem (c). Homework Equations Set theory. Union. Integers. Prime numbers. The Attempt at a Solution I see how we have all multiples of all prime numbers in...

The numbers $a$ and $b$ are prime and satisfy $\dfrac{a}{a+1}+\dfrac{b+1}{b}=\dfrac{2k}{k+2}$ for some positive integer $k$. Find all possible values of $b-a$.

Guys, please help me figure this out: 1) how to calculate the largest prime less than 300 2) why 35 and 37 are not twin primes? 3) the smallest number divisible by five different primes Any input would be greatly appreciated)

Homework Statement Prove that if you have n+1 integers less than or equal to 2n then at least 2 are relatively prime. The Attempt at a Solution the book say integers but I am pretty sure this will only work in the natural numbers. there are n even numbers between 0 and 2n okay and none of...

just a quick question. why is two prime if its has factors, (1+i) and (1-i)?

Dear fellow learners, Through an extracurricular project I have found a really cool equation to count primes. The equation can evaluate Pi(x)+Pi(√x)/2+Pi(cubedroot(x))/3+...Pi(nthroot(x))/n I have directly proved my equation so I now it will be accurate 100% of the time. Although the...

I've been asked in an exercise to show that the function $f(n)$ which returns the $n$th prime is a primitive recursive function. We've covered the basics of primitive recursion, the primitive recursive schematic notation, addition, multiplication, limited subtraction, bounded products, sums...

Hi everyone, :) Here's a question that I am struggling find the answer. Any nudge in the correct direction would be greatly appreciated. Question: Prove that a prime Artinian ring is simple.

Show that for an odd integer $m\ge 5$, $\displaystyle {m\choose 0} 5^{m-1}-{m\choose 1} 5^{m-2}+{m\choose 2} 5^{m-3}-\cdots+{m\choose m-1} $ is not a prime number.

Is there a name for a prime number whose digits are all prime? The first several that I can think of are 2,3,5,7 and 23, 23 being the first double digit prime whose digits are all prime.

this is the program which i wrote: #include<iostream.h> #include<conio.h> #include<stdlib.h> void prime(int p) { if(p==0||p==1) { cout<<"neither prime nor composite"<<endl; getch(); exit(1); } for(int i=2;i<p/2;i++) { if(p%i==0) { cout<<"composite"<<endl; break; } else...

Homework Statement For integers a,b, and c, if a and c are relatively prime and c|ab, then c|b. Knowing that: For any integers p and q, there are integers s and t such that gcd(p,q) = sp + tq. The hint I'm given is that I should form an equation from the fact that they are "relatively...

I am reading Dummit and Foote Section 15.4: Localization. On page 710, D&F make the following statement: ------------------------------------------------------------------------------- "In general, suppose R is a commutative ring. If P is a prime ideal in R[x] then $$ P \cap R $$ is a prime...

I am reading Dummit and Foote, Section 15.4: Localization and am currently working on Proposition 38, part 3 (contraction bijection) - see attachments. I am hoping that someone can demonstrate a proof of the following propostion (without - as D&F do - referring to or relying on translating the...

Hi! :) I have to show that the language $L=\{a^{k},\text{ k is a prime }\}$ is not context-free..I thought that I could show this,using the pumping lemma.I took the word $s^{p}$,and said that if we add $i|vy|$ at the length of $s$,it must still belong in $L$..To show that it is not possible,I...

I talked with an old friend of mine. We discussed prime numbers and Ulams Spiral, and the mathematical patterns that surround us all. He brought up something called the Zeta-Function and something about -1 1/2 and how this all related to prime numbers. I did a google search and found some...

An easy question. All "odd" number can be expressed as a sum of consecutive natural numbers. Example: 35=17+18 35=5+6+7+8+9 35=2+3+4+5+6+7+8Question: Demonstrate that prime numbers (except for the "2"), can only be expressed as the sum of two consecutive natural numbers.

I am reading R.Y. Sharp: Steps in Commutative Algebra, Chapter 3: Prime Ideals and Maximal Ideals. Exerise 3.47 on page 52 reads as follows: ===================================================== Let P be a prime ideal of the commutative ring R. Show that $$ \sqrt (P^n) = P $$...

I understand that when the quark theory was being developed that SU(3) was used to explain the mesons that were ultimately found to be composed of the up, down, and strange quarks. I also get that the SU(3) is grouped as an octet and a singlet, with the eta prime meson being the singlet. But I'm...

trust me this is trivial... As a kid I had a teacher fond of asking if numbers were prime. Of course at the time I had no calculator and did not have many primes remembered. I did not even know the less than square root. I came up with a method that made a simple chart of smaller than the...

I don't know if such thread has been created, all I can find out is one mentioning Zhang's initial bound of $7 \times 10^7$. This has been greatly improved by now so I thought it is worthwhile to post it here as well as the resources which I somehow collected from here and there. History; a...

2 is the "oddest prime of all." Regarding the old humorous "math joke" that 2 is the only even prime, thus it is the "oddest" prime of all. I have a bone to pick with this. I don't think the idea of "even" numbers is any more special than numbers that are divisible by 3 or 5, or anything...

I am reading R.Y.Sharp's book: "Steps in Commutative Algebra. In Chapter 3: Prime Ideals and Maximal Ideals, Exercise 3.22 (ii) reads as follows: ------------------------------------------------------------------------- Determine all the prime ideals of the ring K[X], where K is a field...

Homework Statement use Eular's formula to find the greatest prime number under : If I wasn't forced to use this method I would set up a program to loop through checking for primes Homework Equations F(n) = n^2 + n + 41(0 to 39) or depending on your PoV f(n) = n^2 - n + 41(1 to...

Example (2) on page 682 of Dummit and Foote reads as follows: ------------------------------------------------------------------------ (2) For any field k, the ideal (x) in k[x,y] is primary since it is a prime ideal. ... ... etc...

Example (2) on page 682 of Dummit and Foote reads as follows: ------------------------------------------------------------------------ (2) For any field k, the ideal (x) in k[x,y] is primary since it is a prime ideal. ... ... etc...

Hi, I was wondering what A)the fastest way to find primes is, the fastest I've found so far is the sieve of Eratosthenes. B) The fastest way to find all possible combinations of a set are. e.g. cat-> act,cta,tca,atc,tac,cat any help appreciated. thanks in advance.

I am studying Dummit and Foote Section 15.2. I am trying to understand the proof of Proposition 19 Part (5) on page 682 (see attachment) Proposition 19 Part (5) reads as follows...

I am studying Dummit and Foote Section 15.2. I am trying to understand the proof of Proposition 19 Part (5) on page 682 (see attachment) Proposition 19 Part (5) reads as follows...

I have a problem in understanding the proof to Dummit and Foote Section 15.2, Proposition 19 regarding primary ideals. I hope someone can help. My problem is with Proposition 19 part 4 - but note that part 4 relies on part 2 - see attachment. The relevant sections of Proposition 19 read as...

I have a problem in understanding the proof to Dummit and Foote Section 15.2, Proposition 19 regarding primary ideals. I hope someone can help. My problem is with Proposition 19 part 4 - but note that part 4 relies on part 2 - see attachment. The relevant sections of Proposition 19 read as...

and 2 ^ n + 2 * n? I have checked it for n from 1 to 39. At n = 40, 2 ^ n is over a trillion, and I no longer have the resources to continue checking. I believe this statement is true. As n gets to 40 or more, would this statement become more probable or less probable by the Prime Number...

more and more likely to be true the bigger the even? Primes become more rare, so it seems to me this notion is counter intuitive. :confused: A few recent papers all point to that Goldbach becomes more and more likely the higher up you go. A very large even can be the sum of two large odds or...

Other than the fact that prime numbers are infinite?

Homework Statement If p is a prime and p>3, show that pr\equiv1,5,7 or 11 (mod12) Homework Equations The Attempt at a Solution Do I go about this by knowing that any prime p greater than 3 is of the form 6n+1 or 6n+5? Any direction on how to go about this will be helpful. Thanks.

Homework Statement Im trying to prove that if p is prime, then its square root is irrational. The Attempt at a Solution Is a proof by contradiction a good way to do this? All i can think of is suppose p is prime and √p is a/b, p= (a^2)/ (b^2) Is there any property i can...

Homework Statement Prove or disprove: Is 10^1,000 - 9 Prime? Homework Equations The Attempt at a Solution 10^1,000 = 999...91. Is there a way to logically argue to drop the first nine hundred ninety eight 9's and just look at 91 as being a prime?

I am reading Dummit and Foote Section 13.4 Splitting Fields and Algebraic Closures In particular, I am trying to understand D&F's example on page 541 - namely "Splitting Field of x^p - 2, p a prime - see attached. I follow the example down to the following statement: " ... ... ... so...

prove if the statement is true, else form it's negation and prove that is true: ## \forall y \in (x | x \in \mathbb Z , x \geq 1), 5y^2 + 5y + 1 ## I think it's true, but I can't really even get started to prove it I really suck at these and need help please, thank you!

Homework Statement Assume n = p_1*p_2*p_3*...*p_r = q_1*q_2*q_3*...*q_s, where the p's and q's are primes. We can assume that none of the p's are equal to any of the q's. Why? Homework Equations The Attempt at a Solution I am completely stuck on this. My understanding of the...

It is known that prime numbers become sparser and sparser, with the average distance between one prime number and the next increasing as n approaches infinity. Dividing an even number by 2 results in a bottom half from 1 to n / 2 and a top half from n / 2 to n. For a particular sufficiently...

Hey! From MathWorld on solvable group: But why is that a special case? The way I understand it: the normal series can always be made such that all composition factors are simple, but then the composition factors are both simple and Abelian, and hence (isomorphic to) \mathbb Z_p, i.e. the...

This news is a little dated, but I still found it interesting and wanted to see what everyone else thought about this years discovery of a new "largest" prime: ##2^{(57,885,161)}-1## its 17,425,170 digits long and would span all 7 harry potter books twice. Written out in plain text it would take...

Greetings, humans! (Tongueout) I'm from Ukraine. My English is very bad. So I will use a Google Translate. In 2002, I came up with an interesting piece. I was only 14 years old. I was thinking about fractals and chaos theory, and did not want to learn. Did not want to learn, and were forced to...

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...