What are the factors of 43?

43 has 2 divisors in total: 1, 43. Because 43 is prime, that list can only ever be 1 and the number itself — a prime has no others by definition. They sum to 44.

What is the nearest prime to 43?

The prime immediately below 43 is 41, and the one immediately above is 47. That puts 43 in a gap of 2 below and 4 above.

What is Euler's totient of 43?

φ(43) = 42. For any prime the totient is simply the number minus one, because every smaller positive integer is coprime to it — so all 42 numbers from 1 to 42 share no factor with 43.

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 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.

How does the number of divisors work?

It comes from the exponents in the factorisation, not from the size of the number. Add one to each exponent and multiply: 12 is 2² × 3, so it has (2+1) × (1+1) = 6 divisors. This is why a small number packed with low prime powers can have more divisors than a much larger one — 48 has ten, while 47 has two.

Are there patterns in where primes fall?

Every prime above 3 sits one away from a multiple of 6, which rules out four of every six numbers immediately. Beyond that the sequence is famously irregular: gaps vary unpredictably, and the deepest questions about their distribution, including the Riemann Hypothesis, remain open after more than a century and a half.

Because counting it would break unique factorisation. If 1 were prime then 12 could be written as 2² × 3, or 1 × 2² × 3, or 1⁹ × 2² × 3, and the promise that every number factorises in exactly one way would be false. Excluding 1 keeps that promise, and matches the two-divisor definition.

Is 43 a Prime Number?

43 is the fourteenth prime and the upper half of the twin pair with 41. Together with 37 it bookends a stretch where primes come unusually thickly, before a gap opens on the way to 47. And it is prime: exactly two divisors, 1 and 43, with nothing in between. Proving that takes almost no effort, beca...

YOUR NUMBER

NUMBER

WORTH TRYING

43 IS

PRIME

PRIME GAP TO NEXT

Exactly two divisors, 1 and itself. Previous prime 41, next 47. Twin prime with 41.

Prime factorisation

Number of divisors

Sum of divisors

Totient φ(n)

Previous prime

Next prime

FACTOR BREAKDOWN

PRIME

POWER

CONTRIBUTES

STEPS

Trial-divide 43 by every prime up to √43 ≈ 6 — beyond that a factor would have a partner below it

No divisor found, so 43 has exactly two divisors: 1 and itself

Every prime above 3 sits next to a multiple of 6 — here 43 is 6 × 7 + 1

43 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 43 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: 43 is prime. Nothing needs typing — this page opens the calculator with 43 loaded.
2
Read the STEPS panel to see why. It trial-divides by every prime up to √43 ≈ 6.56, 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 43 a Prime Number?

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