New: Primorial (n#) Added to the Factorial Calculator — Here's What It Does
The factorial calculator's mini-tools row just picked up a fourth calculator: primorial (n#), the product of prime numbers rather than all integers. It sits alongside the existing permutation, combination, and trailing-zero-count tools, with two selectable modes and the same exact BigInt arithmetic used everywhere else on the page. Here's exactly what was added and how to use it.
Up to now, the calculator's "Related calculators" section under the main n! tool covered three companions to factorial: permutation (nPr), combination (nCr), and the trailing-zero count of n!. A fourth card has now joined that row: Primorial (n#), described on the page as "the product of all prime numbers ≤ n (or the first n primes), the 'prime-only' cousin of factorial." It's a small addition, but it closes a gap that came up naturally the last time this tool was covered here — primorial was mentioned as a concept worth having, and now it's actually on the page.
Where exactly does the new primorial tool live?
Short answer: Scroll past the main calculator (the one with the n!, Double, Multi, Sub, Decimal, and Compare tabs) to the "Related calculators" section. The mini-tools grid there now has four cards instead of three: Permutation, Combination, Trailing Zero Count, and the new Primorial card, each with its own inputs and its own "Calculate" button.
Nothing about the six main calculation tabs changed. Primorial doesn't replace or fold into any of them — it's a genuinely separate, self-contained mini-tool with its own input field and output box, exactly the same pattern as the permutation and combination cards next to it.
What are the two primorial modes, and how do they differ?
Short answer: The Primorial card has a Mode dropdown with two options: "Primes ≤ n", which multiplies every prime number up to and including the value you enter, and "First n primes", which instead multiplies exactly n primes counted from the start of the sequence (2, 3, 5, 7, 11, ...), regardless of how large the last one turns out to be.
The distinction matters because the same input number means something different in each mode. Enter n = 13 in "Primes ≤ n" mode, and the tool multiplies every prime that is 13 or smaller: 2 × 3 × 5 × 7 × 11 × 13 = 30,030. Enter the same n = 13 in "First n primes" mode instead, and it multiplies the first thirteen primes in sequence — 2 through 41 — which is a much larger product, since it keeps going well past 13 to gather thirteen primes total.
Illustrative comparison: n = 13 under each primorial mode
Mode
Primes multiplied
Result
Primes ≤ n
2, 3, 5, 7, 11, 13
30,030
First n primes
2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41
304,250,263,527,210
Both results are exact, whole-number products computed with BigInt, the same arbitrary-precision integer type used throughout the rest of the calculator, so there's no rounding even as the primorial grows into a large integer for bigger inputs.
How do you actually run a primorial calculation?
Step-by-step, using the "Primes ≤ n" mode as an example:
Scroll to the "Related calculators" section and find the Primorial (n#) card.
Leave the Mode dropdown on "Primes ≤ n" (the default), or switch it to "First n primes" if that's what you want instead.
Type a value into the n field — the card ships with a default of 13, which you can change to any whole number.
Click the card's "Calculate" button.
Read the result in the output box that appears below the button, showing the exact product.
Because the calculation runs the same way as the rest of the page — BigInt multiplication, no server round trip — a primorial for a reasonably sized n resolves instantly, the same speed you'd expect from the permutation or trailing-zero cards right next to it.
Why does having a primorial calculator next to factorial actually matter?
Short answer: Because primorial and factorial are easy to conflate but behave very differently — primorial only multiplies prime numbers, so it grows far more slowly than a factorial of the same n, and it shows up in number-theory contexts (like arguments about the infinitude of primes) that ordinary factorial doesn't touch.
The gap in growth rate is visible even at small n: 10# (the primorial) is 2 × 3 × 5 × 7 = 210, while 10! (the factorial) is 3,628,800 — more than seventeen thousand times larger, purely because factorial multiplies in every integer from 1 to 10 instead of just the four primes among them. Having both calculations available side by side on the same page makes that difference concrete rather than something you have to compute by hand to appreciate.
Practical note: If you only need the primorial for a small, specific n, working it out by hand (list the relevant primes, multiply them) is entirely feasible. The tool's value shows up once n grows large enough that finding every prime up to it, or the first n primes, by hand becomes tedious or error-prone — the same reason the rest of this page exists for factorial in the first place.
Try the new primorial mini-tool yourself — plus exact n! up to 10000!, double/multifactorial, subfactorial, decimal (Gamma) factorial, permutations, and combinations, all free and instant?
It sits in the "Related calculators" mini-tools row underneath the main factorial calculator, alongside permutation (nPr), combination (nCr), and trailing zero count. It does not replace or change any of the existing six calculation tabs (n!, double, multi, sub, decimal, compare) — it is an additional, separate mini-tool.
What are the two primorial modes and how do they differ?
"Primes ≤ n" multiplies every prime number that is less than or equal to the value you enter, for example n=13 multiplies 2 x 3 x 5 x 7 x 11 x 13. "First n primes" instead multiplies exactly n primes counted from the start of the prime sequence, so n=13 in that mode multiplies the first 13 primes (2 through 41), regardless of how large the last one is.
Is the primorial result exact or approximate?
It is exact. Like every other calculation on this tool, the primorial mini-tool multiplies using JavaScript's BigInt type, so the product of the selected primes is returned as a full, non-rounded integer no matter how many digits it reaches.
Why would I want the product of primes instead of the product of all integers?
Primorial shows up specifically where the question is about prime factors rather than all whole numbers, such as constructing numbers for primality arguments (a classic technique multiplies a set of primes and adds 1), or comparing how much more slowly a prime-only product grows than an ordinary factorial of the same n.
Does adding primorial change any of the calculator's existing factorial results?
No. Primorial is an additive feature: the six factorial calculation modes (n!, double factorial, multifactorial, subfactorial, decimal/Gamma factorial, and compare) and the existing permutation, combination, and trailing-zero mini-tools behave exactly as before. Primorial is a new, independent mini-tool alongside them.
A note on this guide: The example results above (30,030 for 13# under "Primes ≤ n," 210 for 10#, and the "first n primes" product for n=13) were computed by direct multiplication of the stated prime lists and are exact mathematical results, not estimates. This article documents a real feature of the calculator as of the date published and is for general educational reference, not professional or statistical advice.