Counting Prime Numbers Less Than 100,000

It has been determined that there are 9,592 prime numbers below 100,000. Among these, the largest prime number is 99,991.

Understanding Prime Numbers

Prime numbers are fundamental in number theory, consisting of positive integers greater than 1 that have no divisors other than 1 and themselves. Counting the prime numbers in a specific range, such as less than 100,000, involves either a precomputed list or computational methods depending on the size of the range.

Prime Number Counting Algorithms

One of the earliest and most famous algorithms for determining prime numbers is the Sieve of Eratosthenes. This method involves marking the multiples of each prime starting from 2, effectively filtering out all composite numbers. However, for large ranges like 100,000, more advanced algorithms and computational tools are necessary.

Python Code implementation

The following Python code provides a simple and straightforward method to count prime numbers within a specified range:

def count_primes(num):
    if num  2:
        return 0
    primes  [2]
    x  3
    while x  num:
        for y in primes:
            if x % y  0:
                x   2
                break
        else:
            (x)
            x   2
    print(primes)
    return len(primes)

When you run this code with an input of 100,000, it will print out all the prime numbers and return the count as 9,592.

Computational Verification

A more efficient and modern approach involves using computational tools and computer programs. One such program is written in Haskell:

main  print . length . takeWhile ( 100000 $ primes

This program utilizes the primes library to generate a list of all prime numbers less than 100,000 and then prints the length of that list.

Recursive Prime Generation Algorithm

Another method involves a recursive algorithm based on the Sieve of Eratosthenes, but adapted for modern recursive techniques. This algorithm involves constructing an infinite list of infinite lists of multiples, merging them to get composite numbers, and then finding the set difference with the infinite list of natural numbers to get primes. Here's a brief outline of the process:

Construct an infinite list of infinite lists where each list Mp contains the products of the prime number p, starting with pp. Merge all the infinite lists to get a single list of composite numbers. Find the set difference between the composite numbers and the natural numbers to retrieve the primes.

While there are number theoretic methods for estimating the number of primes in a certain range, such as the Prime Number Theorem, they provide approximations rather than exact counts. For precise counts, computational methods like the ones described here are more reliable and efficient.

Conclusion

Counting primes less than 100,000 involves a combination of traditional algorithms, modern computational tools, and recursive methods. Despite the availability of advanced algorithms, the 9,592 prime numbers identified in this range remain a testament to the power and interconnectedness of mathematics and computational science.