What are the factors of 23?

23 has 2 divisors in total: 1, 23. Because 23 is prime, that list can only ever be 1 and the number itself — a prime has no others by definition. They sum to 24.

What is the nearest prime to 23?

The prime immediately below 23 is 19, and the one immediately above is 29. That puts 23 in a gap of 4 below and 6 above.

What is Euler's totient of 23?

φ(23) = 22. For any prime the totient is simply the number minus one, because every smaller positive integer is coprime to it — so all 22 numbers from 1 to 22 share no factor with 23.

Which composites look most like primes?

The odd ones whose smallest factor is 7 or above, because every quick test passes them. 49 is 7², 77 is 7 × 11 and 91 is 7 × 13, and all three read as prime to most people. The lesson is the square-root rule: stopping early because the easy tests passed is precisely how these slip through.

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 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 does the totient function count?

How many numbers below n share no common factor with it. For a prime p the answer is p − 1, because every smaller number is coprime to it. For composites it is smaller, computed as n multiplied by (1 − 1/p) across each distinct prime factor. It is the function at the heart of RSA encryption, which is why a piece of pure number theory from 1763 now protects internet traffic.

What is a Mersenne prime?

A prime of the form 2^p − 1 where p is itself prime. 3, 7, 31 and 127 are the first four. The exponent being prime is necessary but nowhere near sufficient — 2¹¹ − 1 is 2,047, which is 23 × 89. They matter because there is a fast test that works only for numbers of this shape, which is why every record-breaking largest known prime for decades has been a Mersenne prime.

Is 23 a Prime Number?

23 is a prime that opens the first noticeably wide gap in the sequence: the next prime is 29, six further on, and nothing between them survives. It is the ninth prime, and the first for which the gap to the next reaches six. And it is prime: exactly two divisors, 1 and 23, with nothing in between. P...

YOUR NUMBER

NUMBER

WORTH TRYING

23 IS

PRIME

PRIME GAP TO NEXT

Exactly two divisors, 1 and itself. Previous prime 19, next 29.

Prime factorisation

Number of divisors

Sum of divisors

Totient φ(n)

Previous prime

Next prime

FACTOR BREAKDOWN

PRIME

POWER

CONTRIBUTES

STEPS

Trial-divide 23 by every prime up to √23 ≈ 4 — beyond that a factor would have a partner below it

No divisor found, so 23 has exactly two divisors: 1 and itself

Every prime above 3 sits next to a multiple of 6 — here 23 is 6 × 4 − 1

23 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 23 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: 23 is prime. Nothing needs typing — this page opens the calculator with 23 loaded.
2
Read the STEPS panel to see why. It trial-divides by every prime up to √23 ≈ 4.80, 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 23 a Prime Number?

YOUR NUMBER

HOW TO USE

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