What makes a number prime?

Having exactly two divisors: 1 and itself. That is the whole definition, and it is why 1 is not prime — it has only one divisor, not two. The number 2 is prime, and it is the only even one, since every other even number is divisible by 2 as well as by 1 and itself. Composite numbers are everything else above 1, and by the fundamental theorem of arithmetic each of them breaks into primes in exactly one way: 360 is 2³ × 3² × 5 and there is no second answer.

How do I check whether a number is prime by hand?

Divide by each prime in turn, and stop at the square root. To test 97 you need only try 2, 3, 5 and 7, because the square root of 97 is about 9.85. None divides it, so 97 is prime. The reason you can stop there is worth understanding rather than memorising: if a number has a factor above its square root, it must also have a matching factor below it, and you would already have found that one. Checking beyond the square root can only rediscover pairs you have seen.

Why is 1 not a prime number?

Partly by definition, but the definition was chosen for a reason. If 1 counted as prime, the fundamental theorem of arithmetic would collapse: 12 could be written as 2² × 3, or 1 × 2² × 3, or 1⁵ × 2² × 3, and the promise that every number factorises in exactly one way would be false. Excluding 1 keeps that uniqueness. It is also consistent with the two-divisor definition, since 1 has only itself. Mathematicians did once count it as prime, and stopped for precisely this reason.

What is the Sieve of Eratosthenes?

A method from around 240 BC that is still essentially the fastest way to find every prime below a limit. Write the numbers out, take the first uncrossed one — 2 — and strike out all its multiples. Move to the next survivor, 3, and repeat. Each pass can begin at the square of its prime, because everything smaller was already struck by a smaller factor, and the passes stop entirely once the prime exceeds the square root of the limit. Sieving to 100 takes four passes, using 2, 3, 5 and 7, strikes out 74 numbers, and leaves the 25 primes standing.

Why do 91 and 561 catch people out?

91 has no obvious factor — it is odd, does not end in 0 or 5, and its digits do not sum to a multiple of 3 — so the quick divisibility tricks all pass and it feels prime. It is 7 × 13. 561 is a subtler trap: it satisfies the Fermat primality test for every base coprime to it, which means a naive test declares it prime. It is 3 × 11 × 17, and it is the smallest Carmichael number, the class of composites that fool Fermat. The next two are 1105 and 1729 — the same 1729 famous as the taxicab number, which factorises as 7 × 13 × 19. Miller–Rabin, which this page uses, is not fooled by any of them.

What are Euler's totient and the divisor functions?

All three fall out of the factorisation, which is why this page computes them together rather than with separate algorithms. The divisor count multiplies one more than each exponent, so 360 = 2³ × 3² × 5 has 4 × 3 × 2 = 24 divisors. The divisor sum multiplies (p^(a+1) − 1)/(p − 1) across the factors, giving 1,170 for 360. Euler's totient counts how many numbers below n share no factor with it, computed as n times the product of (1 − 1/p); for a prime it is simply p − 1, so φ(97) = 96.

How many primes are there, and how thinly are they spread?

Infinitely many — Euclid proved it around 300 BC with an argument short enough to hold in your head: multiply any finite list of primes together and add 1, and the result is divisible by none of them, so the list was never complete. They do thin out, though. There are 25 primes below 100 and 168 below 1,000, so the density falls from 25% to under 17%. The Prime Number Theorem makes this precise: the count below x approaches x ÷ ln x. At a thousand that estimate is about 16% low, and at a hundred thousand about 10% low, closing slowly.

What are twin primes, Mersenne primes and perfect numbers?

Twin primes are pairs two apart — 3 and 5, 11 and 13, 59 and 61 — and there are 8 such pairs below 100. Whether there are infinitely many is still unproved and is one of the oldest open problems in mathematics. Mersenne primes have the form 2^p − 1 with p itself prime, which is necessary but not sufficient: 2³¹ − 1 = 2,147,483,647 is prime, while 2¹¹ − 1 = 2,047 is 23 × 89. Perfect numbers equal the sum of their proper divisors; 8,128 is one, and every known perfect number pairs with a Mersenne prime.

Why does prime factorisation matter outside mathematics?

Because multiplying two large primes is easy and undoing it is not, and almost all public-key cryptography rests on that asymmetry. RSA takes two primes of several hundred digits each, multiplies them, and publishes the product; recovering the two factors from it is believed to be computationally infeasible at that size, and the security of a great deal of internet traffic depends on that belief. This page factorises 600,851,475,143 into 71 × 839 × 1,471 × 6,857 in a fraction of a second, which sounds impressive until you notice it is a twelve-digit number rather than a six-hundred-digit one.

Prime Number Calculator

One engine for everything you might want to know about a whole number. Hand it a number and it tells you whether that number is prime, and if it is not, exactly what it is made of — the full prime factorisation, how many divisors it has, what they sum to, Euler's totient, the nearest prime on either...

YOUR NUMBER

NUMBER

WORTH TRYING

97 IS

PRIME

PRIME GAP TO NEXT

Exactly two divisors, 1 and itself. Previous prime 89, next 101.

Prime factorisation

Number of divisors

Sum of divisors

Totient φ(n)

Previous prime

Next prime

101

FACTOR BREAKDOWN

PRIME

POWER

CONTRIBUTES

STEPS

Trial-divide 97 by every prime up to √97 ≈ 9 — beyond that a factor would have a partner below it

No divisor found, so 97 has exactly two divisors: 1 and itself

Every prime above 3 sits next to a multiple of 6 — here 97 is 6 × 16 + 1

97 is PRIME

SIEVE OF ERATOSTHENES

Two thousand years old and still the fastest way. Take the first uncrossed number, 2, and strike out every multiple of it. Move to the next survivor, 3, and repeat. Each pass can start at the square of its prime, because anything smaller was already struck by a smaller factor — which is why the passes get shorter and stop entirely once the prime exceeds √100. Whatever survives is prime. Your number 97 is outlined in gold.

HOW MANY PRIMES BELOW x — AND WHAT THE THEORY PREDICTS

4 vs 4

100

25 vs 22

1,000

168 vs 145

10,000

1,229 vs 1,086

100,000

9,592 vs 8,686

Actual count against x ÷ ln x. At x = 10 the estimate overshoots slightly; from 100 upward it runs low, and from 1,000 onward the gap narrows steadily — a ratio of 1.16 at a thousand down to 1.10 at a hundred thousand. That it tends to 1 at all is the Prime Number Theorem, and how slowly it does so is why the proof took a century to find.

Proved, not guessed. Primality here uses a deterministic Miller–Rabin test with a fixed witness set that is proven correct for every number this page accepts, so a result of PRIME is a proof rather than a probability. Ranges are sieved up to 200,000; single numbers are tested up to 4,503,599,627,370,495.

Live simulation · drawn to scale · updates as you type

Created with❤️byeaglecalculator.com

HOW TO USE

1
In TEST mode, type any whole number from 2 up to about four and a half quadrillion. You get the verdict immediately, along with the factorisation if it is composite, the gap to the next prime if it is not, and six derived quantities including the divisor count and Euler's totient.
2
Try the suggested numbers before anything else. 91 looks prime and is 7 × 13. 561 fools the Fermat test but not this one. 8128 is a perfect number, 2147483647 is the Mersenne prime 2³¹ − 1, and 600851475143 factorises into four primes in a fraction of a second.
3
Switch to RANGE to sieve an interval up to 200,000. You get every prime in it, the count, the density as a percentage, the twin pairs and the sum. The quick-range buttons cover the intervals people usually want.
4
Watch the sieve on the right, or drag the scrubber through it a pass at a time. If your number is 100 or below it is outlined in gold on the grid, so you can see whether it survives or gets struck, and by which prime.

WORKED EXAMPLE

Is 97 prime? The square root of 97 is about 9.85, so the only primes worth trying are 2, 3, 5 and 7 — anything larger would need a partner smaller than 9.85, and we are already checking those. 97 is odd, so 2 fails. Its digits sum to 16, which is not a multiple of 3, so 3 fails. It does not end in 0 or 5, so 5 fails. And 7 × 13 = 91, 7 × 14 = 98, so 7 fails too. No divisor found. 97 is prime — the largest two-digit prime, in fact. Its divisor count is 2, its divisor sum is 98, and φ(97) = 96, since every one of the 96 numbers below it is coprime to it. The previous prime is 89 and the next is 101, so 97 sits in a gap of 4 either side. Now contrast 91, which most people would also guess is prime. It is odd, its digits sum to 10, and it does not end in 0 or 5 — every quick test passes. But 7 divides it: 91 = 7 × 13. That is why the square-root rule matters. Stopping at 5 because the small tests passed would have given the wrong answer.

THE FORMULAS

Primeexactly two divisors: 1 and n

Trial division limittest primes up to √n only

Fundamental theoremn = p₁^a¹ × p₂^a² × … uniquely

Number of divisorsd(n) = Π (aᵢ + 1)

Sum of divisorsσ(n) = Π (pᵢ^(aᵢ+1) − 1)/(pᵢ − 1)

Euler totientφ(n) = n · Π (1 − 1/pᵢ)

Totient of a primeφ(p) = p − 1

Sieve pass startbegin each pass at p²

Prime Number Theoremπ(x) ≈ x ÷ ln x

Twin primesp and p + 2 both prime

Mersenne prime2^p − 1, with p itself prime

Perfect numberσ(n) − n = n

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Last updated: July 28, 2026 · Formula verified · Deterministic Miller–Rabin, so a result of PRIME is proven rather than probable · Eagle-eyed accuracy for every calculation.

Prime Number Calculator

YOUR NUMBER

HOW TO USE

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