Gilbreath Equation, Gilbreath Polynomials, Upper and Lower Bound for Gilbreath Conjecture

1. Introduction to $\mathit{GC}$

Let the ordered sequence $P=\left(2,3,5,7,11,13,17,\dots \right)=\left(p_1,\dots ,p_n\right)$ formed by prime numbers, and set

$$p_a^b=\left\{\begin{array}{cc}{p}_{a+1}-p_a,\hfill & \mathrm{if}b=1;\hfill \\ |p_{a+1}^{b-1}-p_a^{b-1}|,\hfill & \mathrm{otherwise}\hfill \end{array}\right.$$

(1)

where $b\in [1;n-1]$. Gilbreath conjectured that $p_1^b=1$. It is likely that this conjecture is satisfied by many other sequences of integers, so it is necessary to define the general properties of all sequences that satisfy this conjecture. Let $GC\left(n\right)$ denote the $GC$ computed on the sequence $\left(p_1,\dots ,p_n\right)$, then there is an interesting computational proof of $GC\left(n\right)$ by A. M. Odlyzko [3] for $n<{10}^{13}$.

2. Properties of Gilbreath sequence

Definition 1. 

Let $S=\left(s_1,s_2,s_3,\dots ,s_n\right)$ be a finite sequence of integers and

$$s_a^b=\left\{\begin{array}{cc}{s}_{a+1}-s_a,\hfill & \mathit{if}b=1;\hfill \\ |s_{a+1}^{b-1}-s_a^{b-1}|,\hfill & \mathit{otherwise}\hfill \end{array}\right.$$

(2)

where $b\in [1;n-1]$, then S is called Gilbreath sequence iff $s_1^b=1\forall b$.

For example, let $S=\left(2,3,5,9,11,13,17\right)$ be a sequence of length $n=7$ and Gilbreath triangle of S

$s_1{s}_2{s}_3{s}_4\dots s_{n-3}{s}_{n-2}{s}_{n-1}{s}_n$

$s_1^1{s}_2^1{s}_3^1{s}_4^1\dots s_{n-3}^1{s}_{n-2}^1{s}_{n-1}^1$

…

$s_1^{n-2}{s}_2^{n-2}$

$s_1^{n-1}$

Replacing values gives

$2359111317$

$124224$

$12202$

$1022$

$120$

$12$

1

Let ${\mathbb{G}}_n$ denote the set of all Gilbreath sequences of length n and $\mathbb{G}$ the set of all Gilbreath sequences. In the previous example the first term of every sequence (except for the original sequence S) is equal to 1, then $S\in {\mathbb{G}}_7$.

Lemma 1. 

Let $S=\left(s_1,\dots ,s_n\right)\in {\mathbb{G}}_n$ and $S^{\prime }=\left(s_1,\dots ,s_{n-1}\right)$ be finite sequences of integers, then $S^{\prime }\in {\mathbb{G}}_{n-1}$.

Proof. 

Consider the Gilbreath triangle of S

$s_1{s}_2{s}_3{s}_4\dots s_{n-3}{s}_{n-2}{s}_{n-1}{s}_n$

$s_1^1{s}_2^1{s}_3^1{s}_4^1\dots s_{n-3}^1{s}_{n-2}^1{s}_{n-1}^1$

…

$s_1^{n-2}{s}_2^{n-2}$

$s_1^{n-1}$

where $s_1^1=\dots =s_1^{n-2}=s_1^{n-1}=1$ as a consequence of $S\in {\mathbb{G}}_n$. Removing the last element of each sequence gives

$s_1{s}_2{s}_3{s}_4\dots s_{n-3}{s}_{n-2}{s}_{n-1}$

$s_1^1{s}_2^1{s}_3^1{s}_4^1\dots s_{n-3}^1{s}_{n-2}^1$

…

$s_1^{n-2}$

which is Gilbreath triangle of $S^{\prime }$, $s_1^1=\dots =s_1^{n-2}=1$ as a consequence of $S\in {\mathbb{G}}_n$, then $S^{\prime }\in {\mathbb{G}}_{n-1}$. □

Definition 2. 

Let $S=\left(s_1,\dots ,s_n\right)\in {\mathbb{G}}_n$ and $S^{\prime }=\left(s_1,\dots ,s_n,k\right)$ be finite sequences of integers. Denote with ${\mathbb{K}}_S$ the set of integers k such that $S^{\prime }\in {\mathbb{G}}_{n+1}$.

Gilbreath triangle of S is

$s_1{s}_2{s}_3{s}_4\dots s_{n-3}{s}_{n-2}{s}_{n-1}{s}_n$

$s_1^1{s}_2^1{s}_3^1{s}_4^1\dots s_{n-3}^1{s}_{n-2}^1{s}_{n-1}^1$

…

$s_1^{n-2}{s}_2^{n-2}$

$s_1^{n-1}$

where $s_1^1=\dots =s_1^{n-2}=s_1^{n-1}=1$ as a consequence of $S\in {\mathbb{G}}_n$. Gilbreath triangle of $S^{\prime }$ is

$s_1{s}_2{s}_3{s}_4\dots s_{n-3}{s}_{n-2}{s}_{n-1}{s}_{n}k$

$s_1^1{s}_2^1{s}_3^1{s}_4^1\dots s_{n-3}^1{s}_{n-2}^1{s}_{n-1}^1|s_n-k|$

…

$s_1^{n-2}{s}_2^{n-2}|s_3^{n-3}-|s_4^{n-4}-|\dots -|s_{n-1}^1-|s_n-k\left|\right|\dots \left|\right||$

$s_1^{n-1}|s_2^{n-2}-|s_3^{n-3}-|s_4^{n-4}-|\dots -|s_{n-1}^1-|s_n-k\left|\right|\dots \left|\right|\left|\right|$

$|s_1^{n-1}-|s_2^{n-2}-|s_3^{n-3}-|s_4^{n-4}-|\dots -|s_{n-1}^1-|s_n-k\left|\right|\dots \left|\right|\left|\right||$

where $s_1^1=\dots =s_1^{n-2}=s_1^{n-1}=1$ as a consequence of $S\in {\mathbb{G}}_n$. If $s_1^n=|s_1^{n-1}-|s_2^{n-2}-|s_3^{n-3}-|s_4^{n-4}-|\dots -|s_{n-1}^1-|s_n-k\left|\right|\dots \left|\right|\left|\right||=1$, then $S^{\prime }\in {\mathbb{G}}_{n+1}$.

Consider the equation

$$|s_1^{n-1}-|s_2^{n-2}-|s_3^{n-3}-|s_4^{n-4}-|\dots -|s_{n-1}^1-|s_n-k\left|\right|\dots \left|\right|\left|\right||=1$$

(3)

We will refer to equation (3) as Gilbreath equation of S. There are $2^n$ values of k that satisfy (3), then ${\mathbb{K}}_S=\{k_1,\dots ,k_{2^n}\}$ is the set of all solutions for k:

$$k_{1,\dots ,2^n}=\pm s_1^{n-1}\pm s_2^{n-2}\pm s_3^{n-3}\pm s_4^{n-4}\pm \dots \pm s_{n-1}^1+s_n\pm 1$$

(4)

The largest value of k that solves (3) is $max{\mathbb{K}}_S=s_1^{n-1}+s_2^{n-2}+s_3^{n-3}+s_4^{n-4}+\dots +s_{n-1}^1+s_n+1$ and the smallest value is $min{\mathbb{K}}_S=-s_1^{n-1}-s_2^{n-2}-s_3^{n-3}-s_4^{n-4}-\dots -s_{n-1}^1+s_n-1=2s_n-max{\mathbb{K}}_S$. A remarkable relation is

$$max{\mathbb{K}}_S+min{\mathbb{K}}_S=2s_n$$

(5)

Lemma 2. 

Let $S=\left(s_1,\dots ,s_n\right)\in {\mathbb{G}}_n$ be a finite sequence of integers where $s_1\in 2\mathbb{Z}$, then $s_2,\dots ,s_n\in 2\mathbb{Z}+1$.

Proof. 

Let $S_1=\left(s_1\right)$, where $s_1\in 2\mathbb{Z}$. From definition 2, $S_2=\left(s_1,k\right)\in {\mathbb{G}}_2$ if $k=s_1\pm 1\in 2\mathbb{Z}+1$. Let now the sequence $S_3=\left(s_1,s_1\pm 1,k\right)$, from definition 2, $S_3\in {\mathbb{G}}_3$ if $k=|\pm 1|+\left(s_1\pm 1\right)\pm 1=1+s_1\pm 1\pm 1$. From the previous step, $s_1\pm 1\in 2\mathbb{Z}+1$. Then $1+s_1\pm 1\in 2\mathbb{Z}$ and $1+s_1\pm 1\pm 1\in 2\mathbb{Z}+1$. By induction, this can be proved for every element of S. If $S\in {\mathbb{G}}_n$ and the first element of S is an even number, then all the other numbers of the sequence will be odd. □

Lemma 3. 

Let $S=\left(s_1,\dots ,s_n\right)\in {\mathbb{G}}_n$ be a finite sequence of integers where $s_1\in 2\mathbb{Z}+1$, then $s_2,\dots ,s_n\in 2\mathbb{Z}$.

Proof. 

Same argument as lemma 3. □

Lemma 4. 

Let $2\mathbb{Z}+\left(\frac{1}{2}\pm \frac{1}{2}\right)$ denote the sets $2\mathbb{Z}$ and $2\mathbb{Z}+1$ and let a finite sequence of integers $S=\left(s_1,\dots ,s_n\right)\in {\mathbb{G}}_n$ where $s_1\in 2\mathbb{Z}+\left(\frac{1}{2}\pm \frac{1}{2}\right)$. Then $s_2,\dots ,s_n\in 2\mathbb{Z}+\left(\frac{1}{2}\mp \frac{1}{2}\right)$.

Proof. 

Lemma 2 and lemma 3. □

Lemma 5. 

Let $S=\left(s_1,\dots ,s_n\right)\in {\mathbb{G}}_n$ and $S^{\prime }=\left(s_1,\dots ,s_n,k\right)\in {\mathbb{G}}_{n+1}$ be finite sequences of integers where $s_1\in 2\mathbb{Z}+\left(\frac{1}{2}\pm \frac{1}{2}\right)$. Then

$k\in {\mathbb{K}}_S=\{x\in \left[min{\mathbb{K}}_S,max{\mathbb{K}}_S\right]\wedge x\in 2\mathbb{Z}+\left(\frac{1}{2}\mp \frac{1}{2}\right)\}$.

Proof. 

Definition 2 and lemma 4. □

An important result regarding equation (4) follows from lemma 5. (4) generates $2^n$ solutions for a finite sequence $\left(s_1,\dots ,s_n,k\right)\in {\mathbb{G}}_{n+1}$ where $\left(s_1,\dots ,s_n\right)\in {\mathbb{G}}_n$. From lemma 5, these solutions are only even or only odd if $s_1$ is odd of even respectively. Therefore, the number of distinct solutions generated by (4) is $2^{n-1}$ since solutions are divided between even and odd.

Theorem 1. 

Let $S=\left(s_1,\dots ,s_n\right)\in {\mathbb{G}}_n$ and $S^{\prime }=\left(s_1,\dots ,s_n,k\right)$ be finite sequences of integers, then $k\in {\mathbb{K}}_S\iff S^{\prime }\in {\mathbb{G}}_{n+1}$.

Proof. 

Suppose that $s_1\in 2\mathbb{Z}+\left(\frac{1}{2}\pm \frac{1}{2}\right)$. Prove the right implication first. From definition 2, $k\in \left[min{\mathbb{K}}_S,max{\mathbb{K}}_S\right]$ and from lemma 5, $k\in 2\mathbb{Z}+\left(\frac{1}{2}\mp \frac{1}{2}\right)$. Then $k\in {\mathbb{K}}_S\Rightarrow S^{\prime }\in {\mathbb{G}}_{n+1}$. Prove the left implication by contradiction. Suppose that $S^{\prime }\in {\mathbb{G}}_{n+1}$ but $k\notin {\mathbb{K}}_S$. Then $k\in \{x\notin \left[min{\mathbb{K}}_S,max{\mathbb{K}}_S\right]\vee x\notin 2\mathbb{Z}+\left(\frac{1}{2}\mp \frac{1}{2}\right)\}$. From definition 2 and lemma 5 it is not possible to have $S^{\prime }\in {\mathbb{G}}_{n+1}$ if $k>max{\mathbb{K}}_S\vee k<min{\mathbb{K}}_S\vee k\notin 2\mathbb{Z}+\left(\frac{1}{2}\mp \frac{1}{2}\right)$. Then it is true $k\in {\mathbb{K}}_S\Leftarrow S^{\prime }\in {\mathbb{G}}_{n+1}$. □

Corollary 1. 

Let $S=\left(s_1,\dots ,s_n\right)$ be a finite sequence of integers, then $S\in {\mathbb{G}}_n\iff min{\mathbb{K}}_{\left(s_1,\dots ,s_m\right)}\le s_{m+1}\le max{\mathbb{K}}_{\left(s_1,\dots ,s_m\right)}\forall m\in \left[\left.1,n\right)\right.$.

Proof. 

As a consequence of definition 2 and equation (4), each m-th element of a sequence S must be within the range between the upper and the lower sequences calculated on all the elements prior to the m-th ones. From definition 2 and according to the solution of Gilbreath equation (4), there cannot exist Gilbreath sequence in which the m-th element is larger than $max{\mathbb{K}}_{\left(s_1,\dots ,s_{m-1}\right)}$, since $max{\mathbb{K}}_{\left(s_1,\dots ,s_{m-1}\right)}$ is the maximum value for the m-th element. The same goes for $min{\mathbb{K}}_{\left(s_1,\dots ,s_{m-1}\right)}$, since it is the smallest value for the m-th element. □

3. Upper and lower bound sequence for P

Let now introduce the definition of two notable Gilbreath sequences. Let $S=\left(s_1,\dots ,s_n\right)\in {\mathbb{G}}_n$ be a finite sequence of integers, from (4), any solutions of the Gilbreath equation cannot be greater than $max{\mathbb{K}}_S$, so the sequence $\left(s_1,\dots ,s_n,max{\mathbb{K}}_S\right)\in {\mathbb{G}}_{n+1}$ is the upper bound sequence for S. Let now the new sequence $S^{\prime }=\left(s_1,\dots ,s_n,max{\mathbb{K}}_S\right)$ and its upper bound sequence $\left(s_1,\dots ,s_n,max{\mathbb{K}}_S,max{\mathbb{K}}_{\left(s_1,\dots ,s_n,max{\mathbb{K}}_S\right)}\right)\in {\mathbb{G}}_{n+2}$ and so on. Equally, let a finite sequence of integers $S=\left(s_1,\dots ,s_n\right)$, from (4), any value of k cannot be smaller than $min{\mathbb{K}}_S$ and the new sequence $S^{\prime }=\left(s_1,\dots ,s_n,min{\mathbb{K}}_S,k\right)$ will have the lower limit for $k=min{\mathbb{K}}_{\left(s_1,\dots ,s_n,min{\mathbb{K}}_S\right)}$ and so on. Then, it is possible to introduce the definition of upper bound Gilbreath sequence and lower bound Gilbreath sequence.

Definition 3. 

Let $S=\left(s_1,\dots ,s_n\right)\in {\mathbb{G}}_n$ be a finite sequence of integers. Let denote with $U_S$ the upper bound Gilbreath sequence for S and with $L_S$ the lower bound Gilbreath sequence for S:

$$U_S=\left(u_1,\dots \right)=\left(s_1,\dots ,s_n,max{\mathbb{K}}_{\left(s_1,\dots ,s_n\right)},max{\mathbb{K}}_{\left(s_1,\dots ,s_n,max{\mathbb{K}}_{\left(s_1,\dots ,s_n\right)}\right)},\dots \right)$$

$$L_S=\left(l_1,\dots \right)=\left(s_1,\dots ,s_n,min{\mathbb{K}}_{\left(s_1,\dots ,s_n\right)},min{\mathbb{K}}_{\left(s_1,\dots ,s_n,min{\mathbb{K}}_{\left(s_1,\dots ,s_n\right)}\right)},\dots \right)$$

The following recursive definition holds:

$$u_i=\left\{\begin{array}{cc}{s}_i,\hfill & \mathit{if}i\le n;\hfill \\ max{\mathbb{K}}_{\left(u_1,\dots ,u_{i-1}\right)},\hfill & \mathit{otherwise}\hfill \end{array}\right.$$

and

$$l_i=\left\{\begin{array}{cc}{s}_i,\hfill & \mathit{if}i\le n;\hfill \\ min{\mathbb{K}}_{\left(u_1,\dots ,u_{i-1}\right)},\hfill & \mathit{otherwise}\hfill \end{array}\right.$$

It is also useful to define a notable sub sequence of $U_S$ and $L_S$.

Definition 4. 

Let $S\in {\mathbb{G}}_n$ be a finite sequence of integers and its $U_S$ and $L_S$. Let define ${\tilde{U}}_S$ and ${\tilde{L}}_S$ as follows:

$${\tilde{U}}_S=\left({\tilde{u}}_1,\dots \right)=\left(max{\mathbb{K}}_S,max{\mathbb{K}}_{\left(S,max{\mathbb{K}}_S\right)},\dots \right)$$

$${\tilde{L}}_S=\left({\tilde{l}}_1,\dots \right)=\left(min{\mathbb{K}}_S,min{\mathbb{K}}_{\left(S,min{\mathbb{K}}_S\right)},\dots \right)$$

The following recursive definition holds:

$${\tilde{u}}_i=\left\{\begin{array}{cc}max{\mathbb{K}}_S,\hfill & \mathit{if}i=1;\hfill \\ max{\mathbb{K}}_{\left(S,{\tilde{u}}_1,\dots ,{\tilde{u}}_{i-1}\right)},\hfill & \mathit{otherwise}\hfill \end{array}\right.$$

$${\tilde{l}}_i=\left\{\begin{array}{cc}min{\mathbb{K}}_S,\hfill & \mathit{if}i=1;\hfill \\ min{\mathbb{K}}_{\left(S,{\tilde{l}}_1,\dots ,{\tilde{l}}_{i-1}\right)},\hfill & \mathit{otherwise}\hfill \end{array}\right.$$

From theorem 1, $U_S\in \mathbb{G}$ and $L_S\in \mathbb{G}$, while elements of ${\tilde{U}}_S$ and ${\tilde{L}}_S$ are all even or all odd, then ${\tilde{U}}_S\notin \mathbb{G}$ and ${\tilde{L}}_S\notin \mathbb{G}$.

From definition 3, let $S=\left(s_1\right)$, then $U_S=\left(s_1,s_1+1,s_1+3,\dots ,s_1+2^n-1\right)$, ${\tilde{U}}_S=\left(s_1+1,s_1+3,\dots ,s_1+2^n-1\right)$, $L_S=\left(s_1,s_1-1,s_1-3,\dots ,s_1-2^n+1\right)$ and ${\tilde{L}}_S=\left(s_1-1,s_1-3,\dots ,s_1-2^n+1\right)$. Table 1 shows some examples of Gilbreath sequences and the closed form for ${\tilde{u}}_n$.

4. Gilbreath polynomials

Definition 5. 

Let $P=\left(p_1,\dots ,p_m\right)$ be the ordered sequence of first m prime numbers and let ${\mathcal{P}}_m\left(n\right)$ be a function such that ${\tilde{u}}_n=2^{m+n-1}+{\mathcal{P}}_m\left(n\right)$ where ${\mathcal{P}}_m\left(n\right)=a_{m,0}+a_{m,1}n+\dots +a_{m,k}{n}^k$, then ${\mathcal{P}}_m$ is called m-th Gilbreath polynomial.

Through simple algebra one can prove that for the ordered sequence of first m prime numbers, ${\mathcal{P}}_m\left(n\right)$ are represented in Table 2. This provides important information about sequence A347924 [7] which is the triangle read by rows where row m is the m-th Gilbreath polynomial and column n is the numerator of the coefficient of the k-th degree term. According to Table 2, this sequence contains the integer term of every m-th Gilbreath polynomials. The related sequence A347925 [6] contains the lowest common denominator of m-th Gilbreath polynomial. It is the sequence of denominators of the polynomials in Table 2.

4.1. Relation between Gilbreath polynomials and $GC$

Gilbreath polynomials are closely related to prime numbers and $GC$. Let a finite sequence of integers $S=\left(s_1,\dots ,s_n\right)$, from theorem 1 is true the following. The relationship $s_2=s_1\pm 1$ must be true, otherwise it would not be true that $s_1^1=1$. As a consequence of lemma 5, for all elements subsequent to $s_1$, the absolute difference of two successive elements must be an integer multiple of 2 so as to maintain the absolute difference of two successive elements as an even value. So, if the first element in the sequence is even, the subsequent elements must be odd and if the first element is odd, the subsequent elements must be even.

Let $P=\left(p_1,p_2\right)=\left(2,3\right)\in {\mathbb{G}}_2$ Gilbreath sequence formed by the first two prime numbers. From (5), $min{\mathbb{K}}_{\left(p_1,p_2\right)}\le p_2\le max{\mathbb{K}}_{\left(p_1,p_2\right)}$ and from theorem 1, $\left(p_1,p_2,p_2\right)\in {\mathbb{G}}_3$. By definition of P, $p_n>p_{n-1}$. Since $min{\mathbb{K}}_{\left(p_1,p_2\right)}\le p_2$, it is certainly true that $min{\mathbb{K}}_{\left(p_1,p_2\right)}\le p_3$. The left inequality is proved for $n=3$ and it is easy to prove for every n. The proof of $min{\mathbb{K}}_{\left(p_1,\dots ,p_{n-1}\right)}\le p_n$ is trivial and holds for all prime numbers, hence $p_n⩽max{\mathbb{K}}_{\left(p_1,\dots ,p_{n-1}\right)}\Rightarrow GC\left(n\right)$. Given Gilbreath polynomials in definition 5, $max{\mathbb{K}}_{\left(p_1,\dots ,p_{n-1}\right)}=2^{n-1}+{\mathcal{P}}_{n-1}\left(1\right)$, then

$$p_n-2^{n-1}⩽{\mathcal{P}}_{n-1}\left(1\right)\Rightarrow GC\left(n\right)$$

(6)

Left side of (6)

$$p_n-2^{n-1}⩽{\mathcal{P}}_{n-1}\left(1\right)$$

(7)

consists of Gilbreath polynomial conjecture whose solution implies $GC$. Unfortunately, bounds for $p_n$ are not enough good to prove (7) however this opens the way for a new approach to the $GC$ [1,2,4,5].

5. Conclusions and future work

Theorem 1, about properties of Gilbreath sequence states that if and only if the first element of a finite sequence of integers $\left(s_1,\dots ,s_n\right)\in {\mathbb{G}}_n$ is even or odd, then for any odd or even integer $max{\mathbb{K}}_S\le k\le min{\mathbb{K}}_S$ respectively the sequence $\left(s_1,\dots ,s_n,k\right)\in {\mathbb{G}}_{n+1}$.

Theorem 1 proves equation (6) involving Gilbreath polynomials and equation (7) implies $GC$. Gilbreath polynomials defined in definition 5 introduce a new interesting tool for the study of the properties of prime numbers, in particular we are interested in the matrix of coefficients of Gilbreath polynomials defined as $\mathcal{G}=a_{m,k}$ and a paper on $\mathcal{G}$ will be published in the future.