There are 78,498 prime numbers between 1 and 1,000,000 (1 million).
Prime numbers are whole numbers greater than 1 that are only divisible by 1 and themselves. The task of counting the number of primes within a given range has intrigued mathematicians for centuries. While there isn't a simple formula to directly calculate the number of primes within a specific range, algorithms and computational methods have allowed us to determine the count for large numbers.
The Prime Number Theorem provides an estimate of the number of primes less than a given number x, denoted by π(x). However, this is an approximation. The actual distribution of prime numbers is somewhat irregular.
Finding all prime numbers between 1 and 1 million typically involves using an algorithm like the Sieve of Eratosthenes. This is an efficient method for identifying all prime numbers up to a specified limit.
Here's a brief explanation of how the Sieve of Eratosthenes works:
- Create a list of consecutive integers from 2 through n (where n is the upper limit, in this case, 1,000,000).
- Initially, let p equal 2, the smallest prime number.
- Enumerate the multiples of p by counting in increments of p from 2p to n, and mark them (they will not be prime).
- Find the smallest number in the list greater than p that is not marked. If there was no such number, stop. Otherwise, let p now equal this new number (which is the next prime), and repeat from step 3.
- When the algorithm terminates, all the numbers remaining not marked in the list are all of the prime numbers below n.
Using a computer to perform this calculation quickly confirms that there are 78,498 prime numbers between 1 and 1,000,000.
