Prime Counting Function Calculator
Count Primes Up To N
Calculate π(N) — the number of primes less than or equal to N
Find Nth Prime
Find the Nth prime number
Formula
π(x) = count of primes ≤ x | PNT: π(x) ≈ x/ln(x) | Li(x) ≈ x/(ln(x) - 1)
Frequently Asked Questions
What is the prime counting function π(x)?
π(x) counts the number of prime numbers less than or equal to x. For example, π(10) = 4 because there are 4 primes (2, 3, 5, 7) up to 10. It is one of the most studied functions in number theory.
What is the Prime Number Theorem?
The Prime Number Theorem states that π(x) ≈ x/ln(x) as x → ∞. This means primes become less frequent but never run out. A better approximation is Li(x) = x/(ln(x) - 1). The exact distribution is connected to the Riemann Hypothesis.
How does the Sieve of Eratosthenes work?
Start with all numbers 2 to N marked as prime. For each prime p found, mark all multiples of p (starting from p²) as composite. The remaining unmarked numbers are prime. It is one of the most efficient ways to find all primes up to a moderate limit.
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