Exercise · ZW-PRIME-NUMBER-THEOREM-008 · ★★☆ Intermediate
If \(p_n \sim n \ln n\), what approximate estimate does the theorem give for the 100th prime?
Choose one option.
Show answer
- C. About \(460\)
Explanation
With n = 100 we get 100 ln 100 ≈ 460.5, close to 460. The value 100 ignores the factor ln n, and 46 omits the factor n.
Review the concept: Prime number theorem
The prime number theorem describes the asymptotic behavior of π(x), the number of primes less than or equal to x, as x tends to infinity.
\[\pi(x) \sim \frac{x}{\ln x}\]
More practice problems: prime number theorem
- ZW-PRIME-NUMBER-THEOREM-001 What does π(x) mean in the prime number theorem?
- ZW-PRIME-NUMBER-THEOREM-002 What does the theorem say about the behavior of π(x) as x tends to infinity?
- ZW-PRIME-NUMBER-THEOREM-003 Which symbol denotes that two functions are asymptotically equivalent?
- ZW-PRIME-NUMBER-THEOREM-004 According to the theorem, what does the ratio π(x)/(x/ln x) tend to as x grows?
- ZW-PRIME-NUMBER-THEOREM-005 What is the approximate value of π(100)?
- ZW-PRIME-NUMBER-THEOREM-006 If π(x) ~ x/ln x, what is the approximate estimate of π(1000) using x/ln x?
- ZW-PRIME-NUMBER-THEOREM-007 What is the approximate probability that an integer near x = 1000 is prime, according to the theore…
- ZW-PRIME-NUMBER-THEOREM-009 What is the approximate value of the ratio π(100)/(100/ln 100)?