On Primorial Numbers

A prime number is a natural number having exactly two factors, $1$ and itself [Nar00]. The natural number $1$ is not considered a prime number since it has only one factor ($1$). From now on, $p_{n}$ will denote the $n$th prime number, here $n\geq 1$. The first eleven prime numbers are $2,3,5,7,11,13,17,19,23,29,31$. Although different classes of prime numbers have been defined [Wik], in this paper, we especially concentrate on some prime number classes derived from the notions of admissible $k$-tuple and $k$-constellation [For99, Wikb].

A $k$-tuple $a=(a_{1},a_{2},\ldots,a_{k})$ of natural numbers is admissible if and only if $a_{1}=0$, $a_{i-1}<a_{i}$ for all $i=2,3,\ldots,k$, and there is a prime number $p>3$ such that $(p,p+a_{2},\ldots,p+a_{k})$ is a $k$-tuple of prime numbers, i.e. $p+a_{i}$ is a prime number for all $i=1,2,\ldots,k$. We say that a prime number $p>3$ satisfies an admissible $k$-tuple $a$ (we denote this situation as $[p|a]$) if and only if $(p,p+a_{2},\ldots,p+a_{k})$ is a $k$-tuple of prime numbers. The diameter of an admissible $k$-tuple $a$ is $dia(a)=a_{k}$ while its gap is the maximum difference between two of its consecutive elements, i.e. $gap(a)=max\left\{a_{i}-a_{i-1}\,\middle|\,i=2,3,\ldots,k\right\}$. For example,

1. (1)

   $a=(0,2)$ is an admissible $2$-tuple. If $p=5$, then $5+0=5$ and $5+2=7$, so $(5,7)$ is a $2$-tuple of prime numbers. Clearly, $dia(a)=a_{2}=2$ and $gap(a)=max\{2-0\}=2$.

2. (2)

   $a=(0,2,6)$ is an admissible $3$-tuple. If $p=5$ then $5+0=5$, $5+2=7$, and $5+6=11$, so $(5,7,11)$ is a $3$-tuple of prime numbers. Clearly, $dia(a)=a_{2}=6$ and $gap(a)=max\{2-0,6-2\}=4$.

3. (3)

   $(0,1)$ is not admissible: since $p>3$ is a prime number then $p=2x+1$ (odd number, i.e., not a multiple of $2$). Then $p+1=2x+1+1=2x+2=2(x+1)$ is an even number, therefore it is not a prime number (contradiction).

4. (4)

   $(0,2,4)$ is not admissible $3$-tuple: since $p>3$ then $p=3x+y$ with $y=1,2$ ($p$ is prime so it is not a multiple of $3$). If $y=1$ then $p+2=3x+1+2=3x+3)3(x+1)$ so $p+2$ is a multiple of $3$ then it is not a prime number (contradiction). If $y=2$ then $p+4=3x+2+4=3x+6=3(x+2)$. So $p+4$ is a multiple of $3$ then it is not a prime number (contradiction).

An admissible $k$-tuple $a=(a_{1},a_{2},\ldots,a_{k})$ is $k$-constellation if and only if for any admissible $k$-tuple $b=(b_{1},b_{2},\ldots,b_{k})$ we have that $dia(a)\leq dia(b)$. For example, $(0,2)$, $(0,4)$, and $(0,6)$ are admissible $2$-tuples but only $(0,2)$ is a $2$-constellation ($2\leq 4\leq 6$).

The following is a short list of prime number classes defined in terms of admissible $k$-tuples and $k$-constellations:

1. (1)

   Twin primes. Two prime numbers $p$ and $q$ are twin primes if and only if $q=p+2$. For example, primes $5$ and $7$ are twin primes since they define the prime $2$-tuple $(5,5+2)$ which satisfies the $2$-prime constellation $(0,2)$.

2. (2)

   Cousin primes. Two prime numbers $p$ and $q$ are called cousin primes if and only if $q=p+4$. For example, primes $7$ and $11$ are cousin primes since they define the prime $2$-tuple $(7,7+4)$ which satisfies the admissible $2$-tuple $(0,4)$.

3. (3)

   Sexy primes. Two prime numbers $p$ and $q$ are called sexy primes if and only if $q=p+6$. For example, primes $5$ and $11$ are sexy primes since they define the prime $2$-tuple $(5,5+6)$ which satisfies the admissible $2$-tuple $(0,6)$.

4. (4)

   Sexy primes triplet. Three prime numbers $p$, $q$, and $r$ are called a sexy prime triplet if and only if $q=p+6$ and $r=p+12$. For example, primes $5$, $11$, and $17$ are a sexy prime triplet since they define the prime $3$-tuple $(5,5+6,5+12)$ which satisfies the admissible $3$-tuple $(0,6,12)$.

5. (5)

   Sexy primes quadruplet. Four prime numbers $p,q,r,$ and $s$ are called a sexy prime quadruple if and only if $q=p+6$, $r=p+12$, and $s=p+18$. For example, primes $5$, $11$, $17$, and $23$ are a sexy prime quadruplet since they define the prime $4$-tuple $(5,5+6,5+12,5+18)$ which satisfies the admissible $4$-tuple $(0,6,12,18)$.

Many interesting operations can be defined over prime numbers, for example, we can define the primorial of the $n$th prime number as the product of the first $n\in\mathbb{N}^{+}$ prime numbers [Dub87], i.e., $\#(n)=p_{n}\#=\prod_{i=1}^{n}p_{k}$. This definition is similar to the definition of the factorial function, so it is possible to define the primorial as a recursive function $\#(0)=1$ and $\#(n)=p_{n}\#(n-1)$ for $n\geq 1$. The first six primorials are $1,2,6,30,210,2310$.

Two natural numbers $a,b\in\mathbb{N}$ are coprime or relative-prime numbers if and only if their greatest common divisor is $1$ ($gcd(a,b)=1$). According to this definition, i) $1$ is relative-prime with any other positive natural number, ii) $0$ is not relative-prime with any natural number, and iii) any prime number is relative-prime with any other prime number. Notice that we can define prime numbers in terms of coprime numbers: A natural number $p>1$ is a prime number if and only if $p$ is relative-prime to $q$ for all natural numbers $1\leq q<p$.

Let $n\in\mathbb{N}$, the set of natural numbers less than $n$, is $\mathbb{Z}_{n}=\{0,1,\ldots,n-1\}$. For example, $\mathbb{Z}_{2}=\{0,1\}$ and $\mathbb{Z}_{\#(2)}=\mathbb{Z}_{2*3}=\mathbb{Z}_{6}=\{0,1,3,4,5\}$. Amount the properties hold by $\mathbb{Z}_{n}$, we are especially interested in the structure of the subset of natural numbers that are relative-prime to $\#(n)$ (sometimes called as totative numbers of $\#(n)$). For instance, consider $\mathbb{Z}_{\#(2)}=\mathbb{Z}_{6}$, the subset of $\mathbb{Z}_{6}$ defined by the natural numbers that are relative-prime to $6$ is $\{1,5\}$. It is clear that any prime number $q$ such $p_{n}<q<\#(n)$ will be a relative-prime number to $\#(n)$.

Euler’s totient function [Eul63] counts the natural numbers, in $\mathbb{Z}_{n}$, which are relative-prime to $n$. Take for example $\mathbb{Z}_{\#(3)}=\mathbb{Z}_{30}$, the subset of natural numbers relative-prime to $30$ is $\{1,7,11,13,17,19,23,29\}$, therefore $\varphi(30)=8$. Euler’s totient function is a multiplicative function, i.e., $\varphi(ab)=\varphi(a)\varphi(b)$ for two coprime numbers $a$ and $b$. Moreover, for a prime number $p$, we have that $\mathbb{Z}_{p}-\{0\}$ is the set of totative numbers of $p$, therefore $\varphi(p)=p-1$. Using these two properties of Euler’s totient function, we can easily compute it on primorial numbers: $\varphi(\#(n))=\prod_{i=1}^{n}(p_{k}-1)=\prod_{i=1}^{n}\varphi(p_{k})$.

We define a $n$-primorial set by ’replacing’ numbers $0$ and $1$ in $\mathbb{Z}_{\#(n)}$, with numbers $\#(n)$ and $\#(n)+1$, respectively. We do this since the number $1$ is always coprime to another positive natural number. Let $n\in\mathbb{N}$, the $n$-primorial set is defined as $\mathbb{Z}^{\#}_{n}=\{2,3,\ldots,\#(n),\#(n)+1\}$. The following is the list of the first five $n$-primorial sets:

1. (1)

   $\mathbb{Z}^{\#}_{1}=\{2,3\}$

2. (2)

   $\mathbb{Z}^{\#}_{2}=\{2,3,4,5,6,7\}$

3. (3)

   $\mathbb{Z}^{\#}_{3}=\{2,3,\ldots,30,31\}$

4. (4)

   $\mathbb{Z}^{\#}_{4}=\{2,3,\ldots,210,211\}$

5. (5)

   $\mathbb{Z}^{\#}_{5}=\{2,3,\ldots,2310,2311\}$

Let $n\in\mathbb{N}$, the $n$-primorial interval is defined as $\mathcal{I}_{n}=\left\{m\,\middle|\,\#(n-1)+1\leq x\leq\#(n)+1\right\}$. The following is the list of the first five $n$-primorial intervals:

1. (1)

   $\mathcal{I}_{1}=\{2,3\}$

2. (2)

   $\mathcal{I}_{2}=\{3,4,5,6,7\}$

3. (3)

   $\mathcal{I}_{3}=\{7,8,\ldots,30,31\}$

4. (4)

   $\mathcal{I}_{4}=\{31,32,\ldots,210,211\}$

5. (5)

   $\mathcal{I}_{5}=\{211,212,\ldots,2310,2311\}$

Notice that primorial intervals are not disjoint sets: the last element of $\mathcal{I}_{n}$ is the first element of $\mathcal{I}_{n+1}$.

We can extend the notion of totatives of $\#(n)$ to set $\mathbb{Z}^{\#}_{n}$ by considering the subset of natural numbers in $\mathbb{Z}^{\#}_{n}$ being relative-prime to $\#(n)$. The following is the list of the first three $n$-totative sets:

1. (1)

   The $1$-totative set is $tot(1)=\{3\}$

2. (2)

   The $2$-totative set is $tot(2)=\{5,7\}$

3. (3)

   The $3$-totative set is $tot(3)=\{7,11,13,17,19,23,29,31\}$

Notice that any prime number $p\in\mathbb{Z}^{\#}_{n}$ must be either $p_{i}$ for some $i=1,2,\ldots,n$ or a $n$-totative number, i.e., we can express the Prime Number Theorem in terms of primorials and $n$-totative numbers. We will explore this relationship in a future paper. In this section, we just extend definitions in sections 1.2-1.4 to $n$-totative numbers.

A $k$-tuple $a=(a_{1},a_{2},\ldots,a_{k})$ of natural numbers is $n$-admissible if and only if $a_{1}=0$, $a_{i-1}<a_{i}$ for all $i=2,3,\ldots,k$, and for all $m\geq n$ there is a $m$-totative number $t_{m}$ such that $(t_{m},t_{m}+a_{2},\ldots,t_{m}+a_{k})$ is a $m$-totative $k$-tuple. Notice that a $n$-admissible $k$-tuple is also a $m$-admissible $k$-tuple for all $m>n$. A $n$-totative number $t$ satisfies an $n$-admissible $k$-tuple $a$ if and only if $a[t]=(t,t+a_{2},\ldots,t+a_{k})$ is a $k$-tuple of $n$-totative numbers. The gap of a $k$ tuple $a$ is the maximum difference between two consecutive elements of the $k$-tuple, i.e., $gap(a)=max\left\{a_{i}-a_{i-1}\,\middle|\,i=2,3,\ldots,k\right\}$.

A $n$-admissible $k$-tuple $a=(a_{1},a_{2},\ldots,a_{k})$ is $n$-strong if and only if for all $m\geq n$ we have that $mod(a_{i},p_{m})\neq mod(a_{j},p_{m})$ with $i=1,\ldots,k-1$ and $j=i+1,\ldots,k$. Notice that being $n$-strong implies that $k\leq p_{n}$.

A $n$-admissible $k$-tuple $a=(a_{1},a_{2},\ldots,a_{k})$ is $n$-totative $k$-constellation if and only if for $a_{k}\leq b_{k}$ for any $n$-admissible $k$-tuple $b=(b_{1},b_{2},\ldots,b_{k})$.

The following is a short list of $n$-totative numbers classes:

1. (1)

   Isolated. Let $a=(0,a_{2},\ldots,a_{k})$ a $n$-admissible $k$-tuple, and $t$ a $n$-totative number such that $a[t]$ is a $n$-totative $k$-tuple. $a[t]$ is isolated if and only if both $a[t-a_{2}]$ and $a=[t_{2}]$ are not $n$-totative $k$-tuples.

2. (2)

   $n$-Twin. Two $n$-totative numbers $p$ and $q$ define a $n$-twin couple if and only if $q=p+2$. Notice that a $n$-twin couple satisfies the $n$-strong $n$-totative $2$-constellation $(0,2)$. Moreover, it is an isolated $2$-tuple when $n>2$ since the $3$-tuple $(0,2,4)$ is not a $n$-admissible $3$-tuple: $t$, $t+2$, and $t+4$ all must be coprime to $3$, therefore $t=3s+1$ or $t=3s+2$ for some natural number $s$. If $t=3s+1$ then $t+2=3s+1+2=3(s+1)$, so $t+2$ is not coprime to $3$. Now if $t=3s+2$ then $t+4=3s+2+4=3(s+2)$, so $t+4$ is not coprime to $3$.

3. (3)

   $n$-Cousin. Two $n$-totative numbers $p$ and $q$ are a $n$-cousin couple if and only if $q=p+4$. Notice that a $n$-cousin couple satisfies the $n$-strong $n$-admissible $2$-tuple $(0,4)$. Moreover, it is an isolated $2$-tuple when $n>3$ ($(0,4,8)$ is not a $n$-admissible $3$-tuple, we left the proof to the reader).

4. (4)

   $n$-Sexy. Two $n$-totative numbers $p$ and $q$ are a $n$-sexy couple if and only if $q=p+6$. Notice that a $n$-sexy couple $(p,p+6)$ satisfies the $n$-admissible $2$-tuple $(0,6)$. Moreover, it is isolated if and only if $p-6$ and $p+12$ are not $n$-totative.

5. (5)

   $n$-Sexy triplet. Three $n$-totative numbers $p$, $q$, and $r$ are a $n$-sexy triplet iff $q=p+6$ and $r=p+12$. Notice that a $n$-sexy triplet $(p,p+6,p+12)$ satisfies the strong $n$-admissible $3$-tuple $(0,6,12)$. Moreover, it is isolated if and only if $p-6$ and $p+18$ are not $n$-totative.

6. (6)

   $n$-Sexy primes quadruplet. Four $n$-totative numbers $p$, $q$, $r$, and $s$ are a $n$-sexy quadruplet iff $q=p+6$, $r=p+12$, and $s=p+18$. Notice that any $n$-sexy quadruplet satisfies the $n$-strong $n$-admissible $4$-tuple $(0,6,12,18)$. Moreover, it is an isolated $4$-tuple ($(0,6,12,18,24)$ is a non $n$-admissible $5$-tuple, we left the proof to the reader).

For determining the number of different $n$-totative number classes, we need some technical Theorems about $n$-admissible $k$-truples.

We can now count twin $n$-totatives and compare them against twin primes.

Table 6 shows the relationship between the count of $n$-totative twin couples ($twin(n)$) and the total number of twin prime couples in the $n$-primorial set, i.e., up to $\#(n)+1$ ($twin_{*}(\#(n)+1)$).

We can now count cousin $n$-totatives and compare them against cousin primes.

Table 7 shows the relationship between the count of $n$-totative cousin couples ($cousin(n)$) and the total number of cousin prime couples in the $n$-primorial set, i.e., up to $\#(n)+1$ ($cousin_{*}(\#(n)+1)$).

We can now count sexy $n$-totatives and compare them against sexy primes.