Primes Definition and 298 Threads

The probability that a randomly chosen integer is divisible by a given integer p is 1/p, regardless of whether p is prime. The probability that 2 distinct randomly chosen integers are divisible by the same prime p is 1/p2. I am not sure however whether the probability that 2 distinct randomly...

I have tested the above proposition thru the integer 90, and have found that the proposition holds true. I'm not sure of the method to prove that this would always be true. Any help, criticism, or proof is welcome.

Proof: Suppose that the only prime numbers of the form ## 6k+1 ## are ## p_{1}, p_{2}, ..., p_{n} ##, and let ## N=4p_{1}^{2}p_{2}^{2}\dotsb p_{n}^{2}+3 ##. Since ## N ## is odd, ## N ## is divisible by some prime ## p ##, so ## 4p_{1}^{2}\dotsb p_{n}^{2}\equiv -3\pmod {p} ##. That is, ##...

I would like to know if my answer is correct and if no ,could you correct.But it should be right I hope:

Proof: Suppose for the sake of contradiction that the only primes of the form ## 8k-1 ## are ## p_{1}, p_{2}, ..., p_{n} ## where ## N=16p_{1}^2p_{2}^2\dotsb p_{n}^2-2 ##. Then ## N=(4p_{1}p_{2}\dotsb p_{n})^2-2 ##. Note that there exists at least one odd prime divisor ## p ## of ## N ## such...

Let ## p ## be an odd prime. Then ## 23 ## is a quadratic residue modulo ## p ## if ## (23|p)=1 ##. Applying the Quadratic Reciprocity Law produces: ## (23|p)=(p|23) ## if ## p\equiv 1\pmod {4} ## ## (23|p)=-(p|23) ## if ## p\equiv 3\pmod {4} ##. Now we consider two cases. Case #1: Suppose ##...

im thinking i should just integrate (binominal distribution 1-2000 * prime probability function) and divide by integral of bin. distr. 1-2000. note that I am looking for a novel proof, not just some brute force calculation. (this isn't homework, I am just curious.)

F(n)=##n^2 −n+41## generates primes for all n<41. Questions: (1) Are there polynomials that have longer lists? (2) Is such a list of polynomials finite (yes, no, unknown)? (3) Same questions for quadratic polynomials?

Consider the repunit ## R_{6}=111111 ##. Then ## R_{6}=111111=1\cdot 10^{5}+1\cdot 10^{4}+1\cdot 10^{3}+1\cdot 10^{2}+1\cdot 10^{1}+1\cdot 10^{0} ##. Note that a positive integer ## N=a_{m}10^{m}+\dotsb +a_{2}10^{2}+a_{1}10+a_{0} ## where ## 0\leq a_{k}\leq 9 ## is divisible by ## 7, 11 ##, and...

Proof: Let ## a>5 ## be an integer. Now we consider two cases. Case #1: Suppose ## a ## is even. Then ## a=2n ## for ## n\geq 3 ##. Note that ## a-2=2n-2=2(n-1) ##, so ## a-2 ## is even. Applying Goldbach's conjecture produces: ## 2n-2=p_{1}+p_{2} ## as a sum of two primes ## p_{1} ## and ##...

Proof: Let ## p ## and ## q ## be primes such that ## p-q=3 ##. Now we consider two cases. Case #1: Suppose ## p ## is an even prime. Then ## p=2 ##, because ## 2 ## is the only even prime. Thus ## 2-q=3 ##, so ## q=-1 ##, which contradicts the fact that ## q ## is prime. Case #2: Suppose ## p...

Proof: Suppose ## p ## and ## p+2 ## are twin primes such that ## p>3 ##. Let ## p=2k+1 ## for some ## k\in\mathbb{N} ##. Then we have ## p+(p+2)=2k+1+(2k+1+2)=4k+4=4(k+1)=4m ##, where ## m ## is an integer. Thus, the sum of twin primes ## p ## and ## p+2 ## is divisible by ## 4 ##. Since ##...

Proof: Suppose ## p ## and ## p+2 ## are twin primes. Then we have ## p(p+2)+1=p^2+2p+1=(p+1)^2 ##. Thus, ## (p+1)^2 ## is a perfect square. Therefore, if ## 1 ## is added to a product of twin primes, then a perfect square is always obtained.

Proof: Consider all primes ## p\leq \sqrt{1949} \leq 43 ## and ## q\leq \sqrt{1951} \leq 43 ##. Then we have ## p\nmid 1949 ## and ## q\nmid 1951 ## for all ## p\leq 43 ##. Thus, ## 1949 ## and ## 1951 ## are both primes. By definition, twin primes are two prime numbers whose difference is ## 2...

Proof: Suppose for the sake of contradiction that ## n ## contains at least three prime factors. Let ## n=p_{1} p_{2}\dotsb p_{X} ## for ## x\geq 3 ##. Note that ## n=p_{1} p_{2} p_{3} ##. Then we have ## p_{1}\nleq \sqrt[3]{n} ##, ## p_{2}\nleq \sqrt[3]{n} ## and ## p_{3}\nleq \sqrt[3]{n} ##...

Proof: Suppose ## p\geq q\geq 5 ## and ## p ## and ## q ## are both primes. Note that ## p ## and ## q ## are not divisible by ## 3 ##, so we have ## p^{2}-1\equiv 0 (mod 3) ## and ## q^{2}-1\equiv 0 (mod 3) ##. This means ## 3\mid((p^{2}-1)-(q^{2}-1)) ##, and so ## 3\mid p^{2}-q^{2} ##. Since...

Proof: ## 17=2^4+1 ## Now I'm stuck. How should I find the correct answers to this problem?

Proof: Suppose that there are infinitely many primes of the form n^2-2. Then we have n^2-2=2^2-2=2, n^2-2=3^2-2=7, n^2-2=5^2-2=23, n^2-2=7^2-2=47...

The Riemann Hypothesis, Explained

If I select two integers at random, what is the probability that their sum will be prime?

w = {0,0 | 1,1 | 2,2...} Let x = number of primes up to w+1 Let y = number of primes up to w-1 Now there's an empty prime box in the 0,0 slot not connected to anything. So I let x = p-1 and y = p+1 p = [p0, p1, p2...] Now p0 becomes 1,0/1 It can be either on or off. For the sake of...

Show by contradiction that $$ \sum_{p\in \mathbb{P}}\dfrac{1}{p} =\sum_{p\;\text{prime}}\dfrac{1}{p} $$ diverges. Which famous result is an immediate corollary?

Often I read that the Riemann Hypothesis (RH) is related to prime numbers because of the equivalence on Re(s)>1 of the zeta function and Eurler's product formula , but is it more accurate that the relevance of the RH to primes (or vice-versa) is either that the RH implies formulas for the...

Hi all, I would like to know if in proving the Catalan's conjecture Preda Mihailescu used the infinitude of primes. Best wishes, DaTario

Hi All. Does anybody have a reference, (book, internet site) - besides those books of Paulo Ribenboim - where one can find a compilation of demonstrations of the Euclid's theorem on the infinitude of primes? As a suggestion, if the known proofs are neither too many not too long, it would be nice...

Homework Statement Let ##n## be odd and a composite number, prove that all of its prime is at most ##\frac{n}{3} ## Homework Equations Some theorems might help? Any ##n>1## must have a prime factor if n is composite then there is a prime ##p<√n## such that ##p|n## The Attempt at a Solution...

How do computers evaluate the number of primes below a given integer?

If you have 2 integers n and n+1, it is easy to show that they have no shared prime factors. For example: the prime factors of 9 are (3,3), and the prime factors of 10 are (2,5). Now if we consider 9 and 10 as a pair, we can collect all their prime factors (2,3,3,5) and find the maximum, which...

Homework Statement https://projecteuler.net/problem=58 Homework EquationsThe Attempt at a Solution def prime(N): if N == 1: return False y = int(N**0.5) for i in range(2,y+1): if N%i == 0: return False return True def finder(N): L = len(N)...

Hello everyone. [Firstly, I didn't know if this belongs here or in General; please move if appropriate]. <Moderator's note: moved to GD> I was reading this paper on the AKS primality test (undergraduates can understand it, highly recommended!), and on page 7 the author brings up the story of...

Homework Statement 1.31 Let ##p## be a prime and let ##q## be a prime that divides ##p - 1##. a) Let ##a \epsilon F^*_p## and let ##b = a^{(p - 1)/q}##. Prove that either ##b = 1## or else ##b## has order ##q##. (Recall that order of ##b## is the smallest ##k \ge 1## such that ##b^k = 1## in...

Does there exist a polynomial P(x) with rational coefficients such that for every composite number x, P(x) takes an integer value and for every prime number x, P(x) does not take on an integer value? Can someone please guide me in the right direction? I've tried to consider the roots of the...

Homework Statement I don't understand the lemma. Homework EquationsThe Attempt at a Solution Isn't all prime number not a product of primes? The lemma doesn't make sense to me... Moreover, if m=2, m-1 is smaller than 2, the inequality also doesn't make sense. Please help me

What's the problem with this trivial solution: n --> n'th prime.

Is there a relation between these concepts? We know that as we move down the number line, the primes become less common. This makes sense: as we get more and more numbers, they could be constituted using all the primes that came before early on and less likely to be constituted by all the recent...

At a sufficient resolution, such as mapping every knowing prime gap... Could a fractal equation created to perfectly describe this (at max possible res.): http://techn.ology.net/the-density-plot-of-the-prime-gaps-is-a-fractal/ Likewise clear a path to a nth-dimensional fractal for prime gaps...

I think that we have to get all 2 digit odd numbers that can be expressed as the sum of 2 primes and subtract that from 45, so I think that the answer would be 45-(number of 2 digit integers n that are prime and have n-2 be prime as well)?

I am reading "Introductory Algebraic Number Theory"by Saban Alaca and Kenneth S. Williams ... and am currently focused on Chapter 1: Integral Domains ... I need some help with understanding Example 1.4.1 ... Example 1.4.1 reads as follows: In the above text by Alaca and Williams we read the...

I am reading "Introductory Algebraic Number Theory"by Saban Alaca and Kenneth S. Williams ... and am currently focused on Chapter 1: Integral Domains ... I need some help with understanding Example 1.4.1 ... Example 1.4.1 reads as follows: In the above text by Alaca and Williams we read the...

I am reading "Introductory Algebraic Number Theory"by Saban Alaca and Kenneth S. Williams ... and am currently focused on Chapter 1: Integral Domains ... I need some help with the proof of Theorem 1.2.2 ... Theorem 1.2.2 reads as follows: https://www.physicsforums.com/attachments/6515 In the...

I am reading "Introductory Algebraic Number Theory"by Saban Alaca and Kenneth S. Williams ... and am currently focused on Chapter 1: Integral Domains ... I need some help with the proof of Theorem 1.2.2 ... Theorem 1.2.2 reads as follows: In the above text from Alaca and Williams, we read the...

Homework Statement Prove that there are infinitely many primes using Mersenne Primes, or show that it cannot be proven with Mersenne Primes. Homework Equations A Mersenne prime has the form: M = 2k - 1 The Attempt at a Solution Lemma: If k is a prime, then M = 2k - 1 is a prime. Proof of...

Show that there exists 2016 consecutive numbers that contains exactly 100 primes.

given : A=B$\times $C with the following characters (1) A,B,C $\in N$ (2)A is a palindrome (3)B and C are all palindromic primes (4) 1000<A<2000 (5) B is a 2-digit number (6) C is a 3-digit number find A

I can prove the twin prime counting function has this form: \pi_2(n)=f(n)+\pi(n)+\pi(n+2)-n-1, where \pi_2(n) is the twin prime counting function, f(n) is the number of twin composites less than or equal to n and \pi(n) is the prime counting function. At n=p_n, this becomes \pi_2(p_n) =...

As far as I can tell for the equation n!/n^2=x or n-1!/n=x, if x is a natural number then it seems n is composite. If x is a non-natural number then it is prime (excluding 4). I am aware that this is not very practical since I am using factorials and the numbers get very large. But it still...

Bear with me, I know nothing. Eons ego @micromass told me about this beautiful formula: \frac {\pi^2} 6 = \prod\limits_{P}\left( 1-\frac 1 {p^2}\right) ^{-1} where p are primes. Just a few minutes ago I have learned about the Basel problem and its solution: \sum \limits_{n=1}^{\infty} \frac...

I am trying to understand a condition for a nonincreasing sequence to converge when summed over its prime indices. The claim is that, given a_n a nonincreasing sequence of positive numbers, then \sum_{p}a_p converges if and only if \sum_{n=2}^{\infty}\frac{a_n}{\log(n)} converges. I have tried...

To start let's use this trick to find a sequence of primes 11+13-7=17 This trick starts with prime 11 and the next in sequence 13. After you add them together to get 24 you subtract 7 which is the prime in sequence before 11, and you will get the answer 17. Now this will find every single prime...

Homework Statement sn= 1/n if n is a prime number; sn = 0 if n is not prime. Homework EquationsThe Attempt at a Solution We know that there are infinitely many prime numbers as n tends to infinity so why is ## \frac{1}{(prime number)} ## not equal to zero when n is a prime number?