Prime numbers are of the outmost importance in pure mathematics, but they are also essential in...

 

Transcribed Image Text:

Prime numbers are of the outmost importance in pure mathematics, but they are also essential in cryptography, a discipline whose modern applications include electronic commerce, chip-based payment cards, digital currencies, computer passwords, and military communications. In all these applications, it is better to work with high prime numbers. Nevertheless, these numbers are rare, as the celebrated Prime Number Theorem reveals: T(n) when n +, (PNT) n log(n) where 7(n) is the number of prime numbers not greater than n. Explain why (PNT) implies that high prime numbers are rare. Prime numbers are of the outmost importance in pure mathematics, but they are also essential in cryptography, a discipline whose modern applications include electronic commerce, chip-based payment cards, digital currencies, computer passwords, and military communications. In all these applications, it is better to work with high prime numbers. Nevertheless, these numbers are rare, as the celebrated Prime Number Theorem reveals: T(n) when n +, (PNT) n log(n) where 7(n) is the number of prime numbers not greater than n. Explain why (PNT) implies that high prime numbers are rare.