Prime Number Checker

Is n prime? Trial division up to √n; identifies smallest divisor and nearest primes if composite.

Inputs

Result

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General calculation reads

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Scientific calculator (Casio fx-991EX)
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How to use this calculator

  • Enter integer.
  • Read result + nearest primes.

About this calculator

A prime is a positive integer >1 with exactly two divisors: 1 and itself. The 6k±1 trial-division algorithm tests divisors up to √n: every prime >3 is of form 6k±1, so we skip 2/3 of integers. For n up to 10^15, this finishes in milliseconds. Beyond that, probabilistic tests (Miller-Rabin) are used. Famous primes: 2 (only even), 7 (Mersenne), 25-million-digit primes for crypto.

Frequently asked

Is 1 prime?+
No — by convention, primes are integers ≥2 with exactly two divisors. 1 has only itself.
Is 0 prime?+
No — 0 has every positive integer as a divisor.
Why test up to √n?+
If n has a factor f > √n, then n/f is a factor < √n. So checking up to √n catches all composites.
Why 6k±1 optimization?+
Every integer is 6k, 6k±1, 6k±2, or 6k+3. The 6k, 6k±2, 6k+3 forms are divisible by 2 or 3 — skip them. Only 6k±1 needs testing.
How many primes?+
Infinite (Euclid). About n/ln(n) primes ≤ n by prime number theorem.

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