No. 1 has only a single divisor (itself), not the two required, so it is neither prime nor composite. 0 is also neither — it is divisible by every integer.

Yes, and it is the only even prime. Every other even number is divisible by 2 in addition to 1 and itself, giving at least three divisors, so it is composite. That makes 2 the smallest prime and the only even one.

Why does the check only test divisors up to the square root?

If n has a divisor larger than {n}, it must pair with a divisor smaller than {n}, because their product is n. So if no prime up to {n} divides n, none larger does either, and n is prime. Testing to the square root is enough.

Is 417 a prime number?

No. {417} ≈ 20.42, and testing the primes up to 20 finds that 417 ÷ 3 = 139 exactly. Since 3 divides it, 417 is composite: 417 = 3 × 139.

How are very large numbers checked?

Numbers beyond a trillion (10^12) are too large to factor quickly by trial division, so the calculator uses the deterministic Miller–Rabin primality test. It returns a definite prime-or-composite verdict for every value up to the 15-digit limit.

Prime Number Calculator

How to use this calculator

Type any whole number from 0 up to 15 digits. The calculator tells you whether it is prime, composite, or neither, colour-codes the verdict, and — for a composite — shows its smallest prime factor. Underneath you get the square-root working and the nearest prime on each side.

Tip: A quick shortcut: if a number ends in 0, 2, 4, 6, or 8 it is divisible by 2, and if its digits add up to a multiple of 3 it is divisible by 3 — so only 2 and 3 themselves are prime among such numbers.

What is a prime number?

A prime number is a whole number greater than 1 with exactly two distinct divisors: 1 and the number itself. 7 is prime because nothing between 1 and 7 divides it evenly. A number greater than 1 that has more than two divisors is composite — for instance $12 = 2 \times 6 = 3 \times 4$ , so 12 has the divisors 1, 2, 3, 4, 6, and 12.

Two whole numbers sit outside this split entirely: 0 and 1 are neither prime nor composite. 1 has only one divisor, so it fails the "exactly two" rule, and 0 is divisible by every integer.

Why 1 is not prime

It is tempting to call 1 prime, but the definition needs two different divisors. 1 divided by 1 is 1 — that is a single divisor, not two. Excluding 1 also keeps the Fundamental Theorem of Arithmetic clean: every integer greater than 1 has a unique prime factorization. If 1 counted as prime you could pad any factorization with extra 1s and uniqueness would break.

2 is the only even prime

Every even number is divisible by 2. For 2 itself that gives just the divisors 1 and 2 — exactly two — so 2 is prime. For any larger even number, 2 is a third divisor on top of 1 and the number, so it is automatically composite. That makes 2 the smallest prime and the only even one; every other prime is odd.

The square-root trial-division method

To test whether $n$ is prime, you only need to try dividing by the primes up to $\lfloor\sqrt{n}\rfloor$ :

Compute $\sqrt{n}$ .
List the primes that are less than or equal to $\sqrt{n}$ .
Divide $n$ by each of those primes.
If none divides evenly, $n$ is prime. If one does, $n$ is composite.

Why the square root is the cut-off: divisors come in pairs $a \times b = n$ . If both were larger than $\sqrt{n}$ , their product would exceed $n$ — impossible. So every composite has a factor at or below $\sqrt{n}$ ; finding just one is enough to declare it composite.

Worked example — is 417 prime?

$\sqrt{417} \approx 20.42$ .
Primes up to 20: 2, 3, 5, 7, 11, 13, 17, 19.
$417 \div 2 = 208.5$ (no); $417 \div 3 = 139$ exactly.
3 divides 417, so 417 is composite: $417 = 3 \times 139$ .

Very large numbers use the Miller–Rabin test

Trial division is fast for small numbers but slows down as values grow. For numbers beyond a trillion ( $10^{12}$ ), this calculator switches to the Miller–Rabin primality test — a fast, deterministic check (using fixed witness bases) that gives an exact prime-or-composite answer for every value up to the 15-digit limit, without listing a factor.

The 25 prime numbers below 100

There are exactly 25 primes below 100 (and 168 below 1000):

The 25 primes below 100
2	3	5	7	11
13	17	19	23	29
31	37	41	43	47
53	59	61	67	71
73	79	83	89	97

Prime numbers are the building blocks of every whole number (the Fundamental Theorem of Arithmetic) and the foundation of modern cryptography: multiplying two large primes is easy, but factoring their product back apart is extremely hard.