Yes. 7 has exactly two divisors, 1 and 7, which is the definition of a prime. It is the 4th prime. Checking it takes almost no work: the square root of 7 is about 2.65, so the only candidates are 2, and none of them divides it.

What are the factors of 7?

7 has 2 divisors in total: 1, 7. Because 7 is prime, that list can only ever be 1 and the number itself — a prime has no others by definition. They sum to 8.

What is the nearest prime to 7?

The prime immediately below 7 is 5, and the one immediately above is 11. That puts 7 in a gap of 2 below and 4 above.

What is Euler's totient of 7?

φ(7) = 6. For any prime the totient is simply the number minus one, because every smaller positive integer is coprime to it — so all 6 numbers from 1 to 6 share no factor with 7.

How many primes are there altogether?

Infinitely many, and Euclid proved it around 300 BC with an argument short enough to hold in your head. Take any finite list of primes, multiply them together and add 1. The result leaves a remainder of 1 when divided by every prime on the list, so either it is prime itself or it has a prime factor that was missing. Either way the list was incomplete.

Do primes thin out as numbers get larger?

Yes, steadily. There are 25 primes below 100 and 168 below 1,000, so the density falls from a quarter to under a sixth. The Prime Number Theorem makes it precise: the count below x approaches x divided by the natural logarithm of x. They never stop entirely, but they do get further apart on average.

What is the Sieve of Eratosthenes?

A method from around 240 BC and still essentially the fastest way to list every prime below a limit. Take the first uncrossed number and strike out all its multiples, then move to the next survivor and repeat. Each pass may start at the square of its prime, because everything smaller was already struck, and the passes stop once the prime exceeds the square root of the limit.

What is a twin prime?

A pair of primes two apart, such as 11 and 13, or 29 and 31. There are eight such pairs below 100. Whether there are infinitely many is one of the oldest unsolved problems in mathematics — it is widely believed to be true and has never been proved, though the gap between what is known and what is believed has narrowed considerably since 2013.

What is a perfect number?

A number equal to the sum of its own proper divisors. 6 is the first, since 1 + 2 + 3 = 6, and 28 is the second, since 1 + 2 + 4 + 7 + 14 = 28. They are extraordinarily rare: the next two are 496 and 8,128, and only about fifty are known in total. Every one found so far is even and pairs with a Mersenne prime.

Is 7 a Prime Number?

7 is the largest single-digit prime, and the last prime the sieve needs when sifting the numbers up to 100. Once the striking prime passes 10 there is nothing left to strike below a hundred, which is why sieving that far takes exactly four passes. And it is prime: exactly two divisors, 1 and 7, with...

YOUR NUMBER

NUMBER

WORTH TRYING

7 IS

PRIME

PRIME GAP TO NEXT

Exactly two divisors, 1 and itself. Previous prime 5, next 11. Twin prime with 5.

Prime factorisation

Number of divisors

Sum of divisors

Totient φ(n)

Previous prime

Next prime

FACTOR BREAKDOWN

PRIME

POWER

CONTRIBUTES

STEPS

Trial-divide 7 by every prime up to √7 ≈ 2 — beyond that a factor would have a partner below it

No divisor found, so 7 has exactly two divisors: 1 and itself

Every prime above 3 sits next to a multiple of 6 — here 7 is 6 × 1 + 1

7 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 7 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.

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HOW TO USE

1
The answer is already on screen: 7 is prime. Nothing needs typing — this page opens the calculator with 7 loaded.
2
Read the STEPS panel to see why. It trial-divides by every prime up to √7 ≈ 2.65, finds nothing, and stops — checking further could only rediscover factor pairs already seen.
3
Check the six tiles for the derived quantities — divisor count, divisor sum, Euler's totient and the primes either side. All of them come from the factorisation rather than from separate searches.
4
Type a different number to carry on, or switch to RANGE to sieve a whole interval at once and see how the primes thin out as the numbers grow.

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 29, 2026 · Formula verified · Deterministic Miller–Rabin, so a result of PRIME is proven rather than probable · Eagle-eyed accuracy for every calculation.

Is 7 a Prime Number?

YOUR NUMBER

HOW TO USE

THE FORMULAS

FREQUENTLY ASKED QUESTIONS

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