Contents
How the primorial calculator works
Enter a whole-number limit n. The calculator finds every prime at or below that limit, multiplies each one once, and returns the exact value of n#.
The limit is not a count of primes
A primorial uses n as an upper boundary. Thus 10# contains 2, 3, 5, and 7; it does not contain the first ten primes. Open the factor list below the answer to check exactly which numbers entered the product.
Only primes appear, and each appears once. Composite numbers such as 6 or 10 do not add a factor because their prime building blocks are already represented.
Three useful checks
5# = 30. The eligible primes are 2, 3, and 5, so the product is 2 × 3 × 5.
6# = 30. Increasing the limit from 5 to 6 changes nothing because 6 is composite.
7# = 210. Seven is prime, so the previous result receives one new factor: 30 × 7.
30# = 6,469,693,230. This result contains the ten primes from 2 through 29.
Stepwise growth and exact arithmetic
The value stays flat between consecutive primes and jumps when the boundary reaches the next prime. This is why 29# and 30# are equal, while 31# is 31 times larger.
The base values 0# and 1# are both 1. No primes satisfy the boundary, so mathematics uses the empty product, whose value is 1.
The calculator accepts integers from 0 through 20,000. At the upper limit the answer has 8,602 digits. Grouping spaces make long values readable, but the copy button returns uninterrupted digits.
Questions about primorials
The notation resembles a factorial, but the factor selection follows a different rule.
What does n# mean?
It is the product of all prime numbers less than or equal to n.
Why are 5# and 6# equal?
Six is not prime, so both limits select the same factors: 2, 3, and 5.
Is 10# the product of the first ten primes?
No. Here 10 is a boundary, so 10# uses only primes no greater than 10 and equals 210.
Why does 1# equal 1?
There are no prime factors below 2. The product of an empty set of factors is defined as 1.
Does the calculator round large answers?
No. It uses exact integer arithmetic and preserves every digit up to the accepted limit.
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