# Talk:Primorial

Page contents not supported in other languages.
WikiProject Mathematics (Rated Start-class, Low-priority)

Does anyone know of a reference with details on the sum of inverses of primorials? The sum of inverse integers diverges, sum of inverse primes diverges, sum of inverse factorials converges to e, so it seems possible that the sum of inverse primorials coverges. --Monguin61 09:15, 15 December 2005 (UTC)

I don't know, but common sense would seem to say that it diverges much like the sum of inverse primes does. I will go to the library. Also, I will do some number crunching with Mathematica. Give me a day or two. PrimeFan 19:39, 15 December 2005 (UTC)
I haven't yet gone to the library, but online I've already found a few interesting references on this matter. I recommend Sloane's as a starting point. The Mathworld article on primorial mentions the interesting relation ${\displaystyle \lim _{n\to \infty }p\sharp _{n}^{1 \over p_{n}}=e}$. PrimeFan 22:50, 16 December 2005 (UTC)

It is straightforward to see that the sum of inverse primorials must converge: Say F(n) is the partial sum of inverse factorials up to 1/n!, and P(n) is the partial sum of the first n inverse primorials. For instance, F(4)=1/1!+1/2!+1/3!+1/4!=1+1/2+1/6+1/24, and P(4)=1/2#+1/3#+1/5#+1/7#=1/2+1/6+1/30+1/210. Then, for every n>=1, P(n)<=F(n). Therefore, in the limit n->infinity, we have that P(n) is dominated by F(n). As the sum of all inverse factorials converges, so must the sum of all inverse primorials. —The preceding unsigned comment was added by 89.152.164.152 (talk) 23:12:52, August 19, 2007 (UTC)<!-

- Template:UnsignedIP -->

Yes, it~'s trivial that it converges. http://www.research.att.com/~njas/sequences/A064648 gives the sum 0.7052301717918009651474316828882485137435776391... I don't know whether it has a simple expression with common functions. PrimeHunter 23:50, 19 August 2007 (UTC)

Is also straightforward to show ${\displaystyle \sum _{n=1}^{\infty }{\frac {1}{p_{n}\#}}}$ is rational, using the Cantor series argument, see [1] 78.196.93.135 (talk) 21:38, 19 July 2016 (UTC)

## P or NP

Calculating factorials are not polylogarithmic in speed, does this change for calculating primorials?? Also what is the order of growth for primorials, for factorials it is O(e^(Nlog(N))? Ozone 19:23, 15 March 2006 (UTC)

I wonder if its order of order of is known, because aside from the actual primes of primes, one is presented with the predicament of locating the sequence of primes to be multiplied. When it gets to larger numbers, it becomes extremely difficult to locate primes. -- He Who Is[ Talk ] 03:06, 25 June 2006 (UTC)

## Continuous

Interestingly, n! is drawn as a continuous line in the current version of this image.

--Abdull 09:13, 9 April 2007 (UTC)

Good point. I think we could redraw this with yellow dots and it should still be intelligible. Thoughts, anyone else? PrimeFan 20:10, 10 April 2007 (UTC)
I think the picture is fine, n! can be continuous, n# can't be continuous. -Gesslein 02:33, 11 April 2007 (UTC)
The 'n#' curve is actually a function of the prime itself, not 'n'. So the caption is misleading. Otherwise, the curve would be more aggressive than 'n!'. 187.181.253.25 (talk) 01:49, 10 January 2023 (UTC)
It seems that you confuse n#, the product of the primes not greater than n, with pn#, the product of the n first primes. D.Lazard (talk) 10:28, 10 January 2023 (UTC)

## Primality Testing

The phrase in the article "...play a role in the search for prime numbers..." is quite the understatement. The primorials are the KEY to locating any and all prime numbers. Reference Sokol's conjecture[2]. One can easily convince oneself of the requirement being a prime offset from a primorial as a necessary (but not sufficient) condition for a prime number. --Billymac00 15:11, 2 June 2007 (UTC)

That reference is selfpublished and the only Google hit on "Sokol's conjecture". It appears to fail Wikipedia:Reliable sources and therefore Wikipedia:Verifiability, so I don't think it should be mentioned in the article. PrimeHunter 02:58, 4 June 2007 (UTC)

I APOLOGIZE, I realized this shortly after posting this comment. I have had to come up with the proper conjecture, submitted with the related sequence to Sloane's OEIS June 4th [3]. The correct wording appears there. The sequence needs verified (missing at least 1 term) but has 65 terms so far thru 2358556200. --Billymac00 13:29, 7 June 2007 (UTC)

OEIS currently contains 130331 sequences and the requirements for submissions are low. If something was conjectured there a few days ago by a mathematically unknown person and has not been mentioned elsewhere, then it seems far from being suited for a Wikipedia article. Your conjecture [4] says:
"Each and every prime must be a prime offset (absolute) from either a primorial, or the product of unique primorials. Note that the condition is required but not sufficient for primeness. The offset is less than the candidate".
This is trivially false since 2 and 3 are counter examples. 2 has no prime offset to a product of unique primorials, and the only prime offset for 3 is 3 which is not less than 3. The sequence is also wrong. The 6th term should be 180 instead of 160. PrimeHunter 01:02, 8 June 2007 (UTC)

well, I don't know why the attitude, you may disregard it but the conjecture works, whether I've not phrased it adequately to cover its implementation...one must check to either side of primorial for instance, which catches 3. 2 is the oddball even, doubt one should condemn it for that ...either way, I am not saying anything about changing the article, which is mostly why I stick to Talk to merely speak to what I feel are points of interest ...you can take up the low submission standards with Neil Sloane I suppose ...yes of course 160 is a typo, should be 180. Neil clearly marked the sequence as needing checking and more terms. And yes, I certainly am unknown in math circles...--Billymac00 04:25, 8 June 2007 (UTC)

The purpose of Wikipedia talk pages is to discuss improvements to the article, and Wikipedia:Conflict of interest suggests to post your own work to a talk page for review by other editors and not add it to the article yourself. When you posted here I naturally assumed it was because you wanted the conjecture added to the article, and I reviewed it as editors are supposed to. My review showed it was unreliable and inappropriate for Wikipedia. Sorry the result was not what you hoped for, but I'm here to make a good encyclopedia and not to make you happy. Mathematics requires precision. If a conjecture about primes is trivially false for 2 and 3 then don't claim it has been checked up to 10^5. Sorry to be blunt but such mistakes can make mathematicians dismiss your work very quickly. If you want to be taken seriously then you should review your own work more carefully before publishing it. Maybe you will be offended and say I have bad attitude but I'm actually spending time trying to help you. Nobody else has bothered to comment. It sounds like you think it's true for 3 because 3 = 2+1, but 1 is not considered a prime number. It's trivial with a computer (a fraction of a GHz second with the Sieve of Eratosthenes) to find counter examples above 10^5. The first is 189239. PrimeHunter 14:50, 8 June 2007 (UTC)

Knowing there sounds like no use in further comment, but in closure on my part, my programming does not shown failure at 189239. I never implied 1 as prime, I said an offset (+/-)1 off a primorial is fairly well-known ocurrence of primes (augmenting the conjecture). If my wife can't make me happy, I doubt you can! I appreciate anyone who is truly trying to be helpful ...thanks--Billymac00 18:06, 8 June 2007 (UTC)

I'm not sure whether you mean that you have not tested 189239 (I got that impression from [5]), or that you say it's not a counter example according to your program. According to my 45 GHz second computation, there are 3525 counter examples below 10^9 to your original conjecture. If you modify the conjecture to skip the prime 2 and allow primes within 1 of a product of primorials, then the only of those counter examples you avoid are 2 and 3. There are many small products of primorials and small primes are common, so it doesn't surprise me that your modified conjecture holds for small numbers but fails on many large numbers. See also the law of small numbers. PrimeHunter 19:16, 8 June 2007 (UTC)
User:PrimeHunter's claims are original research and I suggest s/he does not attempt to publish them in this article. Dcoetzee 00:14, 9 June 2007 (UTC)
I completely agree and I have never considered publishing them in the article or suggested that. Have you read the whole discussion? It is part of my argument why Billymac00's conjecture should not be added to the article. Obviously my comments to this false non-notable conjecture shouldn't be added either. PrimeHunter 01:07, 9 June 2007 (UTC)
Woops. I accidentally mixed you up. I meant User:Billymac00's claims are original research, and should not be added to the article, not yours. I was only supporting your position. Dcoetzee 21:09, 9 June 2007 (UTC)
Thanks for the clarification. PrimeHunter 21:30, 9 June 2007 (UTC)
I have discovered an error in my program. It used an incomplete list of primorial products. The modified conjecture appears to have no counter example below 10^9 (I suspect larger counter examples exist but haven't found them). Sorry about the mistake. However, I still consider the conjecture non-notable and original research; unsuitable for Wikipedia. And I see the OEIS editor has removed the false conjecture (where 2 and 3 were trivial counter examples) from [6]. I guess Billymac00 told him about the problems. The sequence is fixed: 160 has been replaced by 180 (and the empty product 1 is now included). Even if the modified conjecture is added after a new OEIS submission, I think OEIS review of comments is too weak to pass WP:OR in this case. And a conjecture is by definition a guess, so all OEIS could show is that a mathematically unknown person has made a guess about something nobody else has apparently discussed (except me here!). It seems well short of notability for Wikipedia, even if there were no OR problems. PrimeHunter 02:19, 10 June 2007 (UTC)

ok, now that we're more copasetic, any interest in the following? When one comes to this page, there is now a drawn-out commentary that detracts from my original intent of sharing useful information. Any chance you'd be willing to archive off the entire thing, and let me re-post a concise note? Better all-around I'd say...oh, and I'm up for any wager you offer that the Conjecture holds up thru some target date or number threshold...I am working with Sloane on rewording the OEIS comment field ...I'd like to say that some of the typos, loose wording won't ever happen again, but being human they will...--Billymac00 15:15, 10 June 2007 (UTC)

I have striken my false statements about finding counter examples. If "sharing useful information" means you want your work added to a Wikipedia article then I think the existing discussion about it should stay here for reference. If you only want to discuss your research on this talk page, then that is not what talk pages are for. They are for discussing work on the article. We can remove the discussion if you don't want to discuss your research at all. PrimeHunter 19:20, 10 June 2007 (UTC)

this is not about my work. It's about recognizing that the primorials are the key to locating EVERY prime (except 2). You've confirmed approx 51 million primes say so, so far. That seems extremely relevant and germane to the article. I guess you're the editor, so I have no say. Returning to where all this began, the statement "...Primorials play a role in the search for prime numbers..." is in my estimation a gross understatement and mis-characterization. A much more accurate statement would be that "primorials are the key to the location of the prime numbers".--Billymac00 20:43, 10 June 2007 (UTC)

A Wikipedia editor is anyone who chooses to make edits. I'm not "the" editor. Your statements are original research which is against Wikipedia content policies. No reliable source has supported them. Let's look at a numerical example with some simplifications. Suppose we want to know whether a prime p near 10^9 satisfies your conjecture. There are around 70 primorial products n below 2p, so p is only a counter example if all 70 values of abs(n-p) are composite. A random number around 10^9 has chance around log(10^9) ~= 1/20.8 of being prime. Let's say for simplicity that the largest primorial in each product n is 13#. Then abs(n-p) never has a factor <= 13 (because each prime <= 13 divides n but not p). A number with no factor below 13 has around 5.2 times better chance of being prime than a random number (more precisely 2/1 * 3/2 * 5/4 * 7/6 * 11/10 * 13/12 = 5.2135...). If we asume the 70 numbers abs(n-p) behave like similar sized random numbers with no factor <=13, then each of them has chance around 5.2/20.8 = 1/4 of being prime, and 3/4 of being composite. The chance all 70 are composite is around (3/4)^70 ~= 1/557,000,000. In view of this it is unsurprising and unremarkable that no counter example has been found below 10^9. It's easy to make conjectures with no small counter example if they are given enough chances to be fulfilled (70 25% chances per prime around 10^9 in this example). Proving true conjectures can be hard or almost impossible, and I see no strong reason to believe your conjecture is true. Based on heuristics each prime appears to have a tiny chance of being a counter example but there are infinitely many primes. If a conjecture hasn't been proved then it's useless to prove other things such as whether a number is composite. And if your conjecture was actually proved (which I think nobody would have a clue how to approach) then I'm not sure it would help in finding primes, compared to other known methods. The Sieve of Eratosthenes can compute all primes below 10^9 in a few GHz seconds. PrimeHunter 22:06, 10 June 2007 (UTC)
No well-known technique for locating primes or testing primality uses primorials in a fundamental way. The connection is not clearly established to a sufficient degree of notability. The statement should not be modified. Dcoetzee 22:32, 10 June 2007 (UTC)

## Two definitions

I'm modifying the article to reflect that OEIS and MathWorld identify two entirely distinct primorial definitions, one the product of the first n primes, and the other the product of the primes <= n. Superm401 - Talk 21:39, 24 May 2008 (UTC)

This is currently done wrong in the article. If ${\displaystyle p_{n}}$ denotes the n'th prime, then ${\displaystyle p_{n}}$# already agrees with the definition of m# as the product of the primes less than or equal to m. The secondary definition should instead define n# as the product of the first n primes. Njerseyguy (talk) 22:14, 1 December 2009 (UTC)
The article may be unclear. Those who use the ${\displaystyle p_{n}}$# notation apparently only allow to replace n by an integer and not to replace ${\displaystyle p_{n}}$ by the nth prime. So according to that definition, ${\displaystyle p_{5}\#=2\times 3\times 5\times 7\times 11=2310.}$ But that cannot be written as 11# = 2310 even though ${\displaystyle p_{5}}$ = 11 if ${\displaystyle p_{5}}$ means the 5th prime.
The second definition places an integer (sometimes restricted to a prime number) before the # character so that definition would say 11# = 2310, and also 12# = 2310 if a composite number is allowed. I use primorials very often (there are probably hundreds at my website) and have never seen the notation n# for the product of the first n primes. Personally I only use the notation n# and always mean the product of primes ≤ n. PrimeHunter (talk) 00:56, 2 December 2009 (UTC)

## Simplify

So I'm up at 2:25AM trying to waste time by going through each article about the counting numbers i.e. 25,26,27 and I come to, I think it was 29. Here I reach an impasse. It is a primorial prime but I have no idea what that is. I go to the primorial article looking for an answer and cannot possibly understand the mathematical gibberish which is supposed to be an explanation. Could someone please explain what this entails in one English sentence at the beginning of the article for the sake of people like me? Lord mortekai (talk) 07:28, 2 November 2009 (UTC)

The primorial of a number is the product of all primes less than or equal to that number. It is similar to the factorial but using the prime numbers only. For example, the primorial of 29 is 2*3*5*7*11*13*17*19*23*29. The notation commonly used for a primorial is to follow the number with a '#' (similar to the '!' for a factorial), e.g. 29#. --mwalimu59 (talk) 19:38, 2 November 2009 (UTC)
To put it simply, the prime numbers are 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101… . By the standard definition, the first primorial is 2, the second is 2×3, the third is 2×3×5, the forth is 2×3×5×7, the fifth is 2×3×5×7×11, the sixth is 2×3×5×7×11×13, the seventh is 2×3×5×7×11×13×17, the eighth is 2×3×5×7×11×13×17×19, the ninth is 2×3×5×7×11×13×17×19×23, the tenth is 2×3×5×7×11×13×17×19×23×29, and so on. Robo37 (talk) 20:45, 3 November 2009 (UTC)
Wikipedia articles often link to other articles explaining a given term. 29 (number)#In mathematics says: "It is the tenth prime number, and also the fourth primorial prime." Note it links directly to primorial prime (which is not a primorial number but a prime next to a primorial number). 29 = 2×3×5−1. PrimeHunter (talk) 23:46, 3 November 2009 (UTC)

## Use for bases

Shouldn't it be mentioned somewhere that primorials are extremely useful for bases? For Base x, if x is a primorial, more fractions terminate with Base x than with any base excluding ones that are multiples of x. For example, with Base 30 more fractions terminate than with any other base apart from Base 60, Base 90, etc. Robo37 (talk) 20:45, 3 November 2009 (UTC)

This is not true. More fractions terminate in base 462 = 2×3×7×11 than in base 30 = 2×3×5. In base 462 the ratio of non-terminating fractions is 1/2 × 2/3 × 6/7 × 10/11 = 20/77 = 0.259740... In base 30 it is 1/2 × 2/3 × 4/5 = 4/15 = 0.2666... However, the primorials are exactly the bases for which more fractions terminate than in any smaller base. If we were to choose a new common base instead of 10 today then I would say a power of two for computer reasons is more important than the ratio of terminating fractions. PrimeHunter (talk) 23:57, 3 November 2009 (UTC)
Okay I can see how I was wrong, but primorials are still the best bases for showing fractions, and, in fact, if I could choose a new common base - I would choose a primorial as in my opinion less terminating fractions is the only property that is any actually use. And these bases do have a more unique property - Base 6 shows every unit fraction from 1/1 to 1/4 as terminating ones and is the only base that has this property excluding ones that are multiples of 6 like base 12 and base 18, Base 30 shows every unit fraction from 1/1 to 1/6 as terminating ones and is the only base that has this property excluding ones that are multiples of 30 like base 60 and base 90, Base 210 shows every unit fraction from 1/1 to 1/10 as terminating ones and is the only base that has this property excluding ones that are multiples of 210 like base 420 and base 630, and so on. In general, Base x, when x is the nth primorial, represents every unit fraction from 1/1 to 1/y, when y is one less than the nth prime; and no other bases do this other than multiples of x. I probably could have put that simpler but you get the idea.
So… do you think it should be mentioned somewhere in this article? Robo37 (talk) 20:50, 4 November 2009 (UTC)
Do you know a reliable source which has found this worth mentioning? Without that it looks like original research. Base 210 or higher has too many digits to be practical for humans. Base 2 is better known for other reasons, and all even bases have at least as many terminating fractions as base 2. So it appears to me that these considerations are only relevant for base 6 and base 30. Base 30 already mentions something about it. PrimeHunter (talk) 23:53, 4 November 2009 (UTC)
There is a mention of a mixed primorial based radix at Mixed radix#Primorial based radix, so if that is notable enough to be mentioned there, maybe it should be mentioned here...
Maybe we could mention normal primorial bases while on the subject.
And I personally don't see the number of symbols needed a major problem, many cultures use a completely different set of symbols than the ones we use today, so there shouldn’t really be a cap on the height of any particular base... and for Base 210 you could, say, use the 62 symbols used with Base 62, with maybe & and @ afterwards, and then construct the remaining 148 out of those 64 by putting shapes or dots around them. Base 60 doesn't use even use the same digits Base 62 does, it represents each digit in decimal notation, so if we use this for higher bases the number of the base is irrelevant.
Okay I'm rambling... no, I'm not saying that I think there should be a new set of symbols made up just for Wikipedia... that's stupid... I just think that there should be some mention of bases in this article. This article is supposed to mention notable properties about primorials, and this Primorial based radix, to me, looks pretty notable. Robo37 (talk) 11:10, 10 November 2009 (UTC)
Sorry I probably could have worded that better... I fail at this kinda thing. So does anyone object to me mentioning this Primorial based radix somewhere in this article? Robo37 (talk) 09:34, 17 November 2009 (UTC)
Primorials might generate huge numbers of regular numbers quickly, but in the end they still succumb to a large proportion of totatives. I'd say 210 is already over the limit. I would prefer superior highly composite numbers, as they have more factors (better than regular numbers in that they terminate after one place, and that you can more easily use divisibiliy tests for high powers of prime factors of the base). So, sexagesimal IMO is even better than trigesimal for fractions.
Using pure digits A-x for sexagesimal renders addition opaque and multiplication must be done in other ways. 6/10 sexagesimal keeps addition and multiplication algorithms, though they need to be generalized. If you're looking for a SHCN that can be used as a decimal-like base, senary or duodecimal are the best choices. The former is also a primorial. Double sharp (talk) 13:50, 26 November 2013 (UTC)

## Incorrect introduction

The article intro says "There are two conflicting definitions that differ in the interpretation of the argument: the first interprets the argument as an index into the sequence of prime numbers (so that the function is strictly increasing), while the second interprets the argument as a bound on the prime numbers to be multiplied (so that the function value at any composite number is the same as at its predecessor). The rest of this article uses the latter interpretation.".

That paragraph speaks of an 'argument' but no notation is given.

The following section, "Definition for Primorial numbers", defines the p_n# notation as the product of the first n primes.

As the article intro gives no notation and is adjacent to the p_n# notation, this implies that when the intro says "The rest of this article uses the latter interpretation" (meaning, "the argument as a bound on the prime numbers to be multiplied"), that this applies to the p_n# notation. But this is not how the p_n# notation works; so the article is inconsistent.

I suggest deleting the paragraph in the intro starting with "There are two conflicting definitions that differ in the interpretation of the argument" and ending with "The rest of this article uses the latter interpretation."; because:

• this paragraph is incorrect; it is not the case that "The rest of this article uses the latter interpretation"
• the article doesn't use English phrasing such as "the n-th primorial" but rather uses the two notations p_n# and n#, which are defined distinctly; so the disambiguation that this parapragh tries to provide is unnecessary

Bayle Shanks (talk) 19:30, 29 February 2016 (UTC)

## Merge proposal from compositorial

At Wikipedia:Articles for deletion/Compositorial, a merge into this article has been suggested. If you have an opinion on this merge, please weigh in there. —David Eppstein (talk) 18:01, 1 April 2016 (UTC)

## Removal of redundant "n#" column in table of primorial numbers

In the table of primorial numbers, there are columns for both "n#" and "p_n#". This is redundant since for each n, you can look up its p_n in the p_n column and then find p_n#. So the same information is given twice, with no additional use. I move that the n# column be removed.

Jamgoodman (talk) 18:04, 22 June 2017 (UTC)

I oppose. Tables like this are nearly always meant to be read one row at a time. Removing the n# column would make it hard for many readers to find n# by combining information in two rows. Composite n will be extra hard since it's not in the p_n column and they must figure out which value to use. PrimeHunter (talk) 20:06, 22 June 2017 (UTC)

## Proposal for showing the apporximate value in scientific notation in the table

I think it would be convenient to have the table display a column containing an easily readable approximate value of the primorials. For example, for 173# the value could be ≈ 1.66×1068 — Preceding unsigned comment added by 2601:600:8880:3806:2c26:71d1:3a9:4059 (talk)

Agreed, but I think there should only be one value listed per primorial, exact for small and approximate for large, e.g. above 20 digits. I work a lot with primorials and even I don't want to see large decimal expansions. If people actually need large primorials for something then they probably just compute them like me. PrimeHunter (talk) 11:04, 2 June 2018 (UTC)

## Primality test and prime sieve made by primorial

Note: respecting the policy to publish my own job, I declare that the following idea is mine. A demonstration of what is claimed can be found on the website. Similar to Wilson's primality test it's possible to achieve the same result without using the huge factorial necessary to perform the test with the smaller primorial. The cornerstone of the work is Euclid's theorem on the gcd of which it's possible to provide an original demonstration applied to the primorial. The Vinc'S primality test (derived from the theorem) can shrink, inside the primorial, the number to carry on for testing. To tell the truth, Wilson's test has the advantage that it doesn't need to know all the prime numbers less than the number we want to test. On the other hand, the Vinc'S test behaves almost like a sieve (like the Eratosthenes one) but it has the advantage of opening the way to new scenarios (factorization?). VincS (talk) 06:11, 15 April 2022 (UTC) — Preceding unsigned comment added by VincS (talkcontribs) 22:22, 14 April 2022 (UTC)