What are the factors of 34?

34 has 4 divisors in total: 1, 2, 17, 34. Its prime factorisation is 2 × 17, and every one of those 4 divisors is built from those primes. The divisor count follows directly from the exponents: (1 + 1) × (1 + 1) = 4. They sum to 54.

What is the nearest prime to 34?

The prime immediately below 34 is 31, and the one immediately above is 37. Since 34 is composite it sits inside the run between them, 5 numbers wide.

What is Euler's totient of 34?

φ(34) = 16. The totient counts how many numbers below 34 share no common factor with it. Computing it from the factorisation 2 × 17 gives 34 × (1 − 1/2) × (1 − 1/17) = 16.

How large can a gap between primes be?

As large as you like. Take any number n and consider the run from n! + 2 up to n! + n: every one of them is composite, because n! + k is divisible by k. That constructs a gap of any length you choose. Gaps do appear early too — there is nothing between 23 and 29, and nothing between 89 and 97.

What makes a number prime?

Having exactly two divisors: 1 and itself. That is the entire definition, and it is why 1 does not qualify — it has only one divisor. Everything above 1 that is not prime is composite, and by the fundamental theorem of arithmetic every composite breaks into primes in exactly one way.

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.

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.

Is 34 a Prime Number?

34 is the product 2 × 17, and a Fibonacci number. Its four divisors make it about as sparse as an even composite can be, since any even number already has 1, 2, half of itself and itself. It is not prime. 34 = 2 × 17, so it has 4 divisors rather than two, and the smallest factor 2 settles the questi...

YOUR NUMBER

NUMBER

WORTH TRYING

34 IS

NOT PRIME

SMALLEST FACTOR

34 = 2 × 17

Prime factorisation

2 × 17

Number of divisors

Sum of divisors

Totient φ(n)

Previous prime

Next prime

FACTOR BREAKDOWN

PRIME

POWER

CONTRIBUTES

STEPS

Trial-divide 34 by every prime up to √34 ≈ 5 — beyond that a factor would have a partner below it

The smallest prime divisor is 2, so 34 is composite

Full factorisation: 2 × 17

34 is NOT 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 34 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: 34 is not prime, because 34 = 2 × 17. Nothing needs typing — this page opens the calculator with 34 loaded.
2
Read the STEPS panel to see the reasoning: the smallest prime divisor of 34 is 2, which is enough to settle it, and the full factorisation 2 × 17 follows.
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 34 a Prime Number?

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HOW TO USE

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