What are the factors of 29?

29 has 2 divisors in total: 1, 29. Because 29 is prime, that list can only ever be 1 and the number itself — a prime has no others by definition. They sum to 30.

What is the nearest prime to 29?

The prime immediately below 29 is 23, and the one immediately above is 31. That puts 29 in a gap of 6 below and 2 above. A gap of 2 above means 29 and 31 are twin primes.

What is Euler's totient of 29?

φ(29) = 28. For any prime the totient is simply the number minus one, because every smaller positive integer is coprime to it — so all 28 numbers from 1 to 28 share no factor with 29.

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 there a formula that produces only primes?

Nothing useful. Euler noticed that n² + n + 41 gives a prime for every n from 0 to 39, which is remarkable, but it fails at n = 40 where the result is 41². No polynomial with integer coefficients can produce primes for every input — it is a theorem, not merely an absence of cleverness. Formulas that do exist are far too slow to compute.

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.

How far do I have to divide before I can stop?

Up to the square root, and no further. If a number had a factor above its square root it would also have a matching factor below it, and you would already have found that one. Testing past the square root can only rediscover pairs you have seen, which is why the work grows far more slowly than the numbers do.

Is 29 a Prime Number?

29 is the tenth prime, and the lower half of the twin pair with 31. It closes the six-wide gap that opened after 23, which is the sort of irregular spacing that makes the distribution of primes hard to pin down and interesting to study. And it is prime: exactly two divisors, 1 and 29, with nothing i...

YOUR NUMBER

NUMBER

WORTH TRYING

29 IS

PRIME

PRIME GAP TO NEXT

Exactly two divisors, 1 and itself. Previous prime 23, next 31. Twin prime with 31.

Prime factorisation

Number of divisors

Sum of divisors

Totient φ(n)

Previous prime

Next prime

FACTOR BREAKDOWN

PRIME

POWER

CONTRIBUTES

STEPS

Trial-divide 29 by every prime up to √29 ≈ 5 — beyond that a factor would have a partner below it

No divisor found, so 29 has exactly two divisors: 1 and itself

Every prime above 3 sits next to a multiple of 6 — here 29 is 6 × 5 − 1

29 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 29 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: 29 is prime. Nothing needs typing — this page opens the calculator with 29 loaded.
2
Read the STEPS panel to see why. It trial-divides by every prime up to √29 ≈ 5.39, 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 29 a Prime Number?

YOUR NUMBER

HOW TO USE

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