Prime Number Checker

What this tool is used for

• Checking whether a number is prime and seeing its factors.
• Working through a number theory exercise.
• Checking a candidate number for a puzzle or a coding challenge.
• Finding the factors of a number that is not prime.
• Seeing the smallest factor, which is what a primality claim actually turns on.

Frequently Asked Questions

How is primality tested?

Trial division restricted to the 6k ± 1 pattern. After ruling out 2 and 3, every prime is one either side of a multiple of six, so the loop tests i and i+2 stepping by 6 and stops at √n. That is a third of the work of testing every odd number, and it is exact — no probabilistic step is involved.

Why does it stop at a trillion?

Because the loop runs to √n, so 10¹² costs at most a million iterations — instant. Beyond that the wait becomes noticeable and JavaScript's exact-integer range (2⁵³) starts to matter. Cryptographic sizes need Miller-Rabin; trial division here is simply faster to be certain with.

Is 1 a prime number?

No, by definition — a prime has exactly two distinct divisors, and 1 has one. This is not pedantry: if 1 were prime, 12 could be written as 2²×3, 1×2²×3, 1²×2²×3 and so on, and the fundamental theorem of arithmetic's UNIQUE factorisation would collapse. 0 and 1 are neither prime nor composite.

What does the prime factorisation show?

The unique multiset of primes whose product is your number, in exponent form — 360 = 2³ × 3² × 5. Uniqueness is the fundamental theorem of arithmetic, and it is why factorisation underpins GCD, LCM and fraction reduction. The factoriser divides out 2 first, then odd candidates only, which halves the search.

How does it find the next prime?

By testing each successive integer until one passes. There is no formula for 'the next prime', and gaps are irregular — 89 to 97 is a gap of 8 while 101 and 103 are twins two apart. The Prime Number Theorem says gaps near n average about ln(n), so the scan stays short even for large inputs.

Common errors and gotchas

• Treating 1 as prime, which by definition it is not.
• Assuming a number with no small factors is prime without a proper test.
• Assuming a checker's silence means prime, when a timeout and a proof look the same.
• Forgetting 2 is the only even prime, so every other even number factors.
• Using the result for cryptographic reasoning, where the numbers involved are far larger.