# Division of a primorial

What remainder is obtained when the smallest prime number greater than n divides n# ? In other words how do we express n# by Euclid's Division Lemma when the divisor is the smallest prime number greater than n ?

• what do you mean by "n#" ? Commented May 30, 2015 at 14:33
• If $n$ is not a prime, $n$# is defined as $p$#, where $p$ is the largest prime not exceeding $n$. Commented May 30, 2015 at 15:07

PARI/GP gives the following output :

? for(k=1,50,p=prod(j=1,k,prime(j));q=prime(k+1);print(k,"  ",q,"   ",component(
Mod(p,q),2)))
1  3   2
2  5   1
3  7   2
4  11   1
5  13   9
6  17   8
7  19   18
8  23   15
9  29   17
10  31   19
11  37   11
12  41   6
13  43   26
14  47   35
15  53   27
16  59   15
17  61   55
18  67   30
19  71   24
20  73   28
21  79   4
22  83   47
23  89   56
24  97   28
25  101   28
26  103   75
27  107   25
28  109   105
29  113   30
30  127   9
31  131   52
32  137   111
33  139   89
34  149   46
35  151   132
36  157   53
37  163   97
38  167   104
39  173   106
40  179   42
41  181   94
42  191   158
43  193   184
44  197   155
45  199   5
46  211   6
47  223   45
48  227   87
49  229   78
50  233   86


The first column of this table is the number of primes in $p$# $\ :=\ 2 \times 3 \times 5 \times 7 \times ... \times p$. The product runs until the $k$-th prime, which is $p$.

The second column gives $q$, which is the smallest prime greater than $p$ (the $k+1$-th prime).

The third column gives the value of $p$# $\ modulo\ q$

I cannot see any pattern in the last column.