5

That is found is larger However , pel than the previous ones was as lar= and 2013_ two other were found but neither large primes found in 20 as the prime found in A...

Question

That is found is larger However , pel than the previous ones was as lar= and 2013_ two other were found but neither large primes found in 20 as the prime found in August 2008. If the largest prime, 900 pa were written out in a typical newsprint size _ it would fill about reason is th of a newspaper. Why search for such large primes? One to test comp requires trillions of calculations and hence can be used in speed and reliability. Also it is important in writing messages Besides, as computer put

that is found is larger However , pel than the previous ones was as lar= and 2013_ two other were found but neither large primes found in 20 as the prime found in August 2008. If the largest prime, 900 pa were written out in a typical newsprint size _ it would fill about reason is th of a newspaper. Why search for such large primes? One to test comp requires trillions of calculations and hence can be used in speed and reliability. Also it is important in writing messages Besides, as computer put "it's like Mount Everest; expert find people climb mountains?" To keep UP on the ongoing race to ger prime, visit the Web site WWW.mersenne.org: section 5.2 EXERCISE/PROBLEM SET EXERCISES 1. How many factors do each of the following numbers have? 4.22 X 3 b.3' X 52 c.5 X 7x 114



Answers

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}

Let f be the function that assigns to each natural number acts the number of primes that our last then or equal toe acts, for example, off of 12 5 Because if I placed 12 right here, there are 12345 numbers less than it, or half of three. It's actually less than or equal to, so three and two would count. That means when I go to do out of eight, if I think about eight as being right here, there are four prime numbers. Less than eight. 10 actually comes out in the same place. So that's also for and then 50 I placed that number. I would have all of these, which turns out to be 15 if you count. Um

Okay, So to find f of a first thing when you just make a list of all the crimes that are less than eight, so that's gonna be too three I and seven. So f eight would be Eagle Teoh. There there is just equal to four. Okay, the 2nd 1 it wanted was f of 10. So all the primes less than 10 or 235 and seven. So f of 10 is also for okay. And the last one is effort 50. He just wants the number of primes that are less than 50. So we're gonna have to three i 7 11 13 17 19 23 29 31. Uh, 37 41 43 47. That's all of them that are less than 50. That's 123456789 10 11 12 13 14 15. So, after 50 would be equal to 15. Okay. Thank you very much.

Okay, so we see that two factors of 7663 is 79 Andi 97.

To to the fifth minus one is a Mersin. Crime since two to the fifth equals 30 to minus one, which equals 31 31 is a prime, Okay?


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