What is the prime number theorem?
Q: What is the prime number theorem?
A: The prime number theorem is a theorem from number theory that explains how prime numbers are distributed across the number range.
Q: Are prime numbers evenly distributed across the number range?
A: No, prime numbers are not distributed evenly across the number range.
Q: What does the prime number theorem formalize?
A: The prime number theorem formalizes the idea that the probability of hitting a prime number between 1 and a given number becomes smaller as numbers grow.
Q: What is the probability of hitting a prime number between 1 and a given number?
A: The probability of hitting a prime number between 1 and a given number is about n/ln(n), where ln(n) is the natural logarithm function.
Q: Is the probability of hitting a prime number with 2n digits greater than the probability of hitting a prime number with n digits?
A: No, the probability of hitting a prime number with 2n digits is about half as likely than with n digits.
Q: Who proved the prime number theorem?
A: Jacques Hadamard and Charles-Jean de La Vallée Poussin proved the prime number theorem in 1896, over a century after Gauss suspected a link between prime numbers and logarithms in 1793.
Q: What is the average gap between consecutive prime numbers among the first N integers?
A: The average gap between consecutive prime numbers among the first N integers is roughly ln(N).
