Skip to content
ENExplore Calculators

Primorial Calculator

Work out primorial instantly with clear inputs, formula shown and shareable results.

About Primorial Calculator

The primorial multiplies all primes up to a limit. It appears in Euclid's proof of the infinitude of primes, where p# + 1 must have a prime factor larger than p.

Formula

Primorial

n# = product of all primes p ≤ n

Worked example

With the default values below, Primorial Calculator returns Primorial up to the limit of 9,699,690. Change any input above and every figure here is recalculated in your browser.

Inputs used

Multiply all primes up toaccepts 2–300
20
Or use the first this many primesaccepts 1–100
5

Results

Primorial up to the limit
9,699,690
Primes multiplied
8
Primes used
2, 3, 5, 7, 11, 13, 17, 19
Primorial of the first n primes
2,310

Step by step

  1. 1Values used
  2. 2Primorial
  3. 3Primorial up to the limit= 9,699,690
  4. 4Primes multiplied= 8
  5. 5Primes used= 2, 3, 5, 7, 11, 13, 17, 19
  6. 6Primorial of the first n primes= 2,310

Assumptions and limits

  • The result is derived only from the 2 inputs listed above. Anything not among them is not part of the calculation.
  • In particular, this calculator does not take inflation, tax, fees or charges or compounding frequency into account.
  • Figures are rounded for display to whole numbers, so totals built from several rounded parts can differ slightly from the unrounded arithmetic.

How these calculators are built and tested is described on the Methodology page. If a figure looks wrong, please report it.

Frequently Asked Questions

What is 20#?

2 × 3 × 5 × 7 × 11 × 13 × 17 × 19 = 9699690.

Why does Euclid's proof use it?

Because p# + 1 leaves remainder 1 when divided by every prime up to p, so its prime factors must be larger — hence no largest prime.

Related Calculators

Primorial Calculator – Calculate Primorial up to the limit