Advance maths question Don't use AI & Answer quickly 183. Explain the significance of prime gaps in the distribution of prime numbers.
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For example, the gap between the primes 3 and 5 is 2, and the gap between 7 and 11 is 4. Show more…
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(a) An integer $N$ is to be selected at random from $\left\{1,2, \ldots,(10)^{3}\right\}$ in the sense that each integer has the same probability of being selected. What is the probability that $N$ will be divisible by $3 ?$ by $5 ?$ by $7 ?$ by $15 ?$ by $105 ?$ How would your answer change if $(10)^{3}$ is replaced by $(10)^{k}$ as $k$ became larger and larger? (b) An important function in number theoryone whose properties can be shown to be related to what is probably the most important unsolved problem of mathematics, the Riemann hypothesis-is the Möbius function $\mu(n),$ defined for all positive integral values $n$ as follows: Factor $n$ into its prime factors. If there is a repeated prime factor, as in $12=$ $2 \cdot 2 \cdot 3$ or $49=7 \cdot 7,$ then $\mu(n)$ is defined to equal 0. Now let $N$ be chosen at random from $\left\{1,2, \ldots(10)^{k}\right\},$ where $k$ is large. Deter$\operatorname{mine} P\{\mu(N)=0\}$ as $k \rightarrow \infty$ Hint: To compute $P\{\mu(N) \neq 0\},$ use the identity $$\prod_{i=1}^{\infty} \frac{P_{i}^{2}-1}{P_{i}^{2}}=\left(\frac{3}{4}\right)\left(\frac{8}{9}\right)\left(\frac{24}{25}\right)\left(\frac{48}{49}\right) \cdots=\frac{6}{\pi^{2}}$$ where $P_{i}$ is the $i$ th-smallest prime. (The number 1 spot a primer
A prime number is a positive integer that has no factors other than 1 and itself. The first few primes are $2,3,5,7,$ $11,13,17, \ldots$ We denote by $\pi(n)$ the number of primes that are less than or equal to $n .$ For instance, $\pi(15)=6$ because there are six primes smaller than $15 .$ \begin{equation} \begin{array}{l}{\text { (a) Calculate the numbers } \pi(25) \text { and } \pi(100) \text { . }} \\ {\text { [Hint: To find } \pi(100) \text { , first compile a list of the primes }} \\ {\text { up to } 100 \text { using the sieve of Eratosthenes: Write the }} \\ {\text { numbers from } 2 \text { to } 100 \text { and cross out all multiples of } 2 \text { . }}\\{\text { Then cross out all multiples of } 3 . \text { The next remaining }} \\ {\text { number is } 5, \text { so cross out all remaining multiples of it, }} \\ {\text { and so on. } .}\end{array} \end{equation} \begin{equation} \begin{array}{l}{\text { (b) By inspecting tables of prime numbers and tables }} \\ {\text { of logarithms, the great mathematician } \mathrm{K} \text { . F. Gauss }} \\ {\text { made the guess in } 1792 \text { (when he was } 15 ) \text { that the num- }} \\ {\text { ber of primes up to } n \text { is approximately } n / \ln n \text { when } n \text { is }} \\ {\text { large. More precisely, he conjectured that }}\end{array} \end{equation} $$\lim _{n \rightarrow \infty} \frac{\pi(n)}{n / \ln n}=1$$ \begin{equation} \begin{array}{l}{\text { This was finally proved, a hundred years later, by }} \\ {\text { Jacques Hadamard and Charles de la Vallee Poussin }} \\ {\text { and is called the Prime Number Theorem. Provide }} \\ {\text { evidence for the truth of this theorem by computing the }} \\ {\text { ratio of } \pi(n) \text { to } n / \ln n \text { for } n=100,1000,10^{4}, 10^{5} \text { , }} \\ {\pi\left(10^{4}\right)=1229, \pi\left(10^{5}\right)=9592, \pi\left(10^{6}\right)=78,498} \\ {\pi\left(10^{7}\right)=664,579 .}\end{array} \end{equation} \begin{equation} \begin{array}{l}{\text { (c) Use the Prime Number Theorem to estimate the number }} \\ {\text { of primes up to a billion. }}\end{array} \end{equation}
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Produce a detailed written explanation of the importance of prime numbers within the field of computing. It is recommended that your explanation do not exceed 700 words.
Madhur L.
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