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Teens can shop, but parents must approve purchases. Amazon also allows you to have packages delivered to your car as long as you have a Buick, Cadillac, Chevrolet, GMC, or Volvo made in or later. The cars also need to have an active OnStar or Volvo On Call account, and the service is available in only 37 cities. Amazon Day: Amazon added this feature to its Prime membership in Amazon Day offers consumers the option to schedule deliveries.

You can still choose to receive orders on other days and by one-day or two-day delivery if an item is needed sooner. This also allows Amazon to consolidate items into fewer boxes and cut down on the number of trips to your home, helping to reduce its carbon footprint.

The company says it aims to make 50 percent of its shipments net-zero carbon by Of course, not everything about Amazon Prime is worth the membership. But take note: At the end of the trial period, your credit card will be charged the membership fee automatically for the next period unless you cancel in advance. Get discounts, product recommendations, and buying advice from the shopping experts at CR. Sign up for our Shopping newsletter. We respect your privacy. All email addresses you provide will be used just for sending this story.

Pros and Cons of Amazon Prime Before signing up, consider how much you'll take advantage of the services. Since this method is based on the idea of sieving sifting the composite numbers, it is called the Sieve of Eratosthenes.

We will demonstrate the sieve of Eratosthenes on the list of prime numbers smaller than , which is hopefully still in front of you Figure 2. Circle the number 2, since it is the first prime number, and then erase all its higher multiples, namely all the composite even numbers. Move on to the next non-erased number, the number 3. Since it was not erased, it is not a product of smaller numbers, and we can circle it knowing that it is prime.

Again, erase all its higher multiples. Notice that some of them, such as 6, have been already deleted, while others, such as 9, will be erased now. The next non-erased number—5—will be circled. Again, erase all its higher multiples: 10, 15, and 20 have already been deleted, but 25 and 35, for instance, should be erased now. Continue in the same manner. Until when? All numbers smaller than that were not erased are prime numbers and can be safely circled! What is the frequency of prime numbers?

How many prime numbers are there, approximately, between 1,, and 1,, one million and one million plus one thousand and how many between 1,,, and 1,,, one billion and one billion plus one thousand?

Can we estimate the number of prime numbers between one trillion 1,,,, and one trillion plus one thousand? Calculations reveal that prime numbers become more and more rare as numbers get larger. But is it possible to state an accurate theorem that will express exactly how rare they are? Such a theorem was first stated as a conjecture by the great mathematician Carl Friedrich Gauss in , at the age of The nineteenth-century mathematician Bernhard Riemann Figure 1 , who influenced the study of prime numbers in modern times more than anyone else, developed further tools needed to deal with it.

But a formal proof of the theorem was given only in , a century after it had been stated. It is interesting to note that both men were born around the time of the death of Riemann.

The precise formulation of the prime number theorem, even more so the details of its proof, require advanced mathematics that we cannot discuss here. But put less precisely, the prime number theorem states that the frequency of prime numbers around x is inversely proportional to the number of digits in x.

Indeed, computer calculations show that there are 75 prime numbers in the first window, 49 in the second and only 37 in the third, between one trillion and one trillion plus one thousand. The same information can be pictured as a graph, shown below Figure 3.

Notice that any time we meet a new prime along the x -axis, the graph rises by 1, so the graph takes the shape of steps Figure 3A. On a small scale, it is hard to detect a pattern in the graph. It is quite easy to prove that we can find arbitrarily large intervals in which there are no prime numbers, meaning spans where the graph does not rise.

On a larger scale, however, the graph looks smooth Figure 3B. This smooth curve seen on a large scale demonstrates the prime number theorem. Systems in probability, such as coin flipping, behave in this way. It is impossible to predict the result of a single coin flipping, but over time, if the coin is unbiased, it will come up heads half the time. What is surprising is that the prime number system is not probabilistic, but it still behaves in many ways as if it were randomly selected.

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In select cities, you can even enable Amazon Key in-home delivery , so you don't even need to be home to bring your packages indoors. It's a secure in-home delivery on millions of eligible items. Eligible shoppers can also get packages delivered to their garage or car when it's parked in a publicly accessible area. Sign up for Insider Reviews' weekly newsletter for more buying advice and great deals. You can purchase syndication rights to this story here. Disclosure: Written and researched by the Insider Reviews team.

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