I was reading the book "Elementary number theory by David Burton" and I came across this paragraph.
Paragraph: "For a prime $p$, define $p^{\#}$ to be the product of all primes that are less than or equal to $p$. Numbers of the form $p^{\#} + 1$ might be termed Euclidean numbers, because they appear in Euclid's scheme for proving the infinitude of primes."
I just want to know is it standard notation to call product of primes less than or equal to $p$ as $p^{\#}$ or is it just the case in the book?
Asked
Active
Viewed 124 times
-1
Bill Dubuque
- 282,220
HARSH KUMAR
- 157
-
2It's not very common but this is the "primorial" and Wikipedia uses a version of this notation: https://en.wikipedia.org/wiki/Primorial – Qiaochu Yuan Jan 19 '25 at 18:28
-
3It is quite common on this site, see for example here, or here, or here etc. Please use MathJax. Here is a tutorial. – Dietrich Burde Jan 19 '25 at 19:08
-
2@DietrichBurde The (somewhat common) notation is not $p^{#}$, it's $p#$. – jjagmath Jan 19 '25 at 19:30
-
1Not sure about the downvotes? – Martin Brandenburg Jan 19 '25 at 21:14
-
@Martin Likely the downvotes are for lack of research effort, e.g. googling "what are common notations for the product of all primes up to p ?" yields the name "primorial" and the common postfix sharp notation, then googling "common notations for primorial" shows no such exponentiated sharp notation. – Bill Dubuque Jan 19 '25 at 23:09
-
@BillDubuque Thank you for the clarification. – Martin Brandenburg Jan 19 '25 at 23:58
-
@DietrichBurde I am wondering if one of these posts is then actually a duplicate target. Also this, for example. – Martin Brandenburg Jan 20 '25 at 00:31
-
Please do not answer in the comments, @QiaochuYuan. – Shaun Jan 20 '25 at 15:21
1 Answers
2
The (somewhat common) notation for the product of all the primes less or equal to $n$ is not $n^{\#}$, but the slightly different $n\#$, as you can see in the links shared in the comments. I don't know why Burton chose to adopt that variant.
jjagmath
- 22,582