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

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Riccardo Gatti^*

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Abstract

Let $S=\left(s_1, \ldots, s_n\right)$ be a finite sequence of integers. Then $S$ is a Gilbreath sequence of length $n$, $S\in\mathbb{G}_n$, iff $s_1$ is even or odd and $s_2, \ldots, s_n$ are respectively odd or even and $\min\mathbb{K}_{\left(s_1, \ldots, s_m\right)}\leq s_{m+1}\leq\max\mathbb{K}_{\left(s_1, \ldots, s_m\right)}\forall m\in\left[\left.1, n\right)\right.$. This, applied to the order sequence of prime number $P$, defines Gilbreath polynomials and two integer sequences A347924 \cite{oeisA347924} and A347925 \cite{oeisA347925} which are used to prove that Gilbreath conjecture $GC$ is implied by $p_n-2^{n-1}\leqslant\mathcal{P}_{n-1}\left(1\right)$ where $\mathcal{P}_{n-1}\left(1\right)$ is the $n-1$-th Gilbreath polynomial at 1.

Keywords:

Subject: Computer Science and Mathematics - Algebra and Number Theory

1. Introduction to $GC$

Let the ordered sequence

P = (2, 3, 5, 7, 11, 13, 17, \dots) = (p_{1}, \dots, p_{n})

formed by prime numbers, and set

p_{a}^{b} = \{\begin{matrix} p_{a + 1} - p_{a}, & if b = 1; \\ | p_{a + 1}^{b - 1} - p_{a}^{b - 1} |, & otherwise \end{matrix}

(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

G C (n)

denote the

G C

computed on the sequence

(p_{1}, \dots, p_{n})

, then there is an interesting computational proof of

G C (n)

by A. M. Odlyzko [3] for

n < 10^{13}

2. Properties of Gilbreath sequence

Definition 1.

Let

S = (s_{1}, s_{2}, s_{3}, \dots, s_{n})

be a finite sequence of integers and

s_{a}^{b} = \{\begin{matrix} s_{a + 1} - s_{a}, & if b = 1; \\ | s_{a + 1}^{b - 1} - s_{a}^{b - 1} |, & otherwise \end{matrix}

(2)

where

b \in [1; n - 1]

, then S is called Gilbreath sequence iff

s_{1}^{b} = 1 \forall b

For example, let

S = (2, 3, 5, 9, 11, 13, 17)

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

2 3 5 9 11 13 17

1 2 4 2 2 4

1 2 2 0 2

1 0 2 2

1 2 0

1 2

Let

G_{n}

denote the set of all Gilbreath sequences of length n and

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 G_{7}

Lemma 1.

Let

S = (s_{1}, \dots, s_{n}) \in G_{n}

and

S^{'} = (s_{1}, \dots, s_{n - 1})

be finite sequences of integers, then

S^{'} \in 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 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^{'}

s_{1}^{1} = \dots = s_{1}^{n - 2} = 1

as a consequence of

S \in G_{n}

, then

S^{'} \in G_{n - 1}

. □

Definition 2.

Let

S = (s_{1}, \dots, s_{n}) \in G_{n}

and

S^{'} = (s_{1}, \dots, s_{n}, k)

be finite sequences of integers. Denote with

K_{S}

the set of integers k such that

S^{'} \in 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 G_{n}

. Gilbreath triangle of

S^{'}

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 | | \dots | | |

s_{1}^{n - 1} | s_{2}^{n - 2} - | s_{3}^{n - 3} - | s_{4}^{n - 4} - | \dots - | s_{n - 1}^{1} - | s_{n} - k | | \dots | | | |

| s_{1}^{n - 1} - | s_{2}^{n - 2} - | s_{3}^{n - 3} - | s_{4}^{n - 4} - | \dots - | s_{n - 1}^{1} - | s_{n} - k | | \dots | | | | |

where

s_{1}^{1} = \dots = s_{1}^{n - 2} = s_{1}^{n - 1} = 1

as a consequence of

S \in 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 | | \dots | | | | | = 1

, then

S^{'} \in 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 | | \dots | | | | | = 1

(3)

We will refer to equation (3) as Gilbreath equation of S. There are

2^{n}

values of k that satisfy (3), then

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 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 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 = 2 s_{n} - max K_{S}

. A remarkable relation is

max K_{S} + min K_{S} = 2 s_{n}

(5)

Lemma 2.

Let

S = (s_{1}, \dots, s_{n}) \in G_{n}

be a finite sequence of integers where

s_{1} \in 2 Z

, then

s_{2}, \dots, s_{n} \in 2 Z + 1

Proof.

Let

S_{1} = (s_{1})

, where

s_{1} \in 2 Z

. From definition 2,

S_{2} = (s_{1}, k) \in G_{2}

k = s_{1} \pm 1 \in 2 Z + 1

. Let now the sequence

S_{3} = (s_{1}, s_{1} \pm 1, k)

, from definition 2,

S_{3} \in G_{3}

k = | \pm 1 | + (s_{1} \pm 1) \pm 1 = 1 + s_{1} \pm 1 \pm 1

. From the previous step,

s_{1} \pm 1 \in 2 Z + 1

. Then

1 + s_{1} \pm 1 \in 2 Z

and

1 + s_{1} \pm 1 \pm 1 \in 2 Z + 1

. By induction, this can be proved for every element of S. If

S \in 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 = (s_{1}, \dots, s_{n}) \in G_{n}

be a finite sequence of integers where

s_{1} \in 2 Z + 1

, then

s_{2}, \dots, s_{n} \in 2 Z

Proof.

Same argument as lemma 3. □

Lemma 4.

Let

2 Z + (\frac{1}{2} \pm \frac{1}{2})

denote the sets

2 Z

and

2 Z + 1

and let a finite sequence of integers

S = (s_{1}, \dots, s_{n}) \in G_{n}

where

s_{1} \in 2 Z + (\frac{1}{2} \pm \frac{1}{2})

. Then

s_{2}, \dots, s_{n} \in 2 Z + (\frac{1}{2} \mp \frac{1}{2})

Proof.

Lemma 2 and lemma 3. □

Lemma 5.

Let

S = (s_{1}, \dots, s_{n}) \in G_{n}

and

S^{'} = (s_{1}, \dots, s_{n}, k) \in G_{n + 1}

be finite sequences of integers where

s_{1} \in 2 Z + (\frac{1}{2} \pm \frac{1}{2})

. Then

k \in K_{S} = {x \in [min K_{S}, max K_{S}] \land x \in 2 Z + (\frac{1}{2} \mp \frac{1}{2})}

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

(s_{1}, \dots, s_{n}, k) \in G_{n + 1}

where

(s_{1}, \dots, s_{n}) \in 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 = (s_{1}, \dots, s_{n}) \in G_{n}

and

S^{'} = (s_{1}, \dots, s_{n}, k)

be finite sequences of integers, then

k \in K_{S} \Leftrightarrow S^{'} \in G_{n + 1}

Proof.

Suppose that

s_{1} \in 2 Z + (\frac{1}{2} \pm \frac{1}{2})

. Prove the right implication first. From definition 2,

k \in [min K_{S}, max K_{S}]

and from lemma 5,

k \in 2 Z + (\frac{1}{2} \mp \frac{1}{2})

. Then

k \in K_{S} \Rightarrow S^{'} \in G_{n + 1}

. Prove the left implication by contradiction. Suppose that

S^{'} \in G_{n + 1}

but

k \notin K_{S}

. Then

k \in {x \notin [min K_{S}, max K_{S}] \lor x \notin 2 Z + (\frac{1}{2} \mp \frac{1}{2})}

. From definition 2 and lemma 5 it is not possible to have

S^{'} \in G_{n + 1}

k > max K_{S} \lor k < min K_{S} \lor k \notin 2 Z + (\frac{1}{2} \mp \frac{1}{2})

. Then it is true

k \in K_{S} \Leftarrow S^{'} \in G_{n + 1}

. □

Corollary 1.

Let

S = (s_{1}, \dots, s_{n})

be a finite sequence of integers, then

S \in G_{n} \Leftrightarrow min K_{(s_{1}, \dots, s_{m})} \leq s_{m + 1} \leq max K_{(s_{1}, \dots, s_{m})} \forall m \in [1, n)

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 K_{(s_{1}, \dots, s_{m - 1})}

, since

max K_{(s_{1}, \dots, s_{m - 1})}

is the maximum value for the m-th element. The same goes for

min K_{(s_{1}, \dots, s_{m - 1})}

, 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 = (s_{1}, \dots, s_{n}) \in G_{n}

be a finite sequence of integers, from (4), any solutions of the Gilbreath equation cannot be greater than

max K_{S}

, so the sequence

(s_{1}, \dots, s_{n}, max K_{S}) \in G_{n + 1}

is the upper bound sequence for S. Let now the new sequence

S^{'} = (s_{1}, \dots, s_{n}, max K_{S})

and its upper bound sequence

(s_{1}, \dots, s_{n}, max K_{S}, max K_{(s_{1}, \dots, s_{n}, max K_{S})}) \in G_{n + 2}

and so on. Equally, let a finite sequence of integers

S = (s_{1}, \dots, s_{n})

, from (4), any value of k cannot be smaller than

min K_{S}

and the new sequence

S^{'} = (s_{1}, \dots, s_{n}, min K_{S}, k)

will have the lower limit for

k = min K_{(s_{1}, \dots, s_{n}, min K_{S})}

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 = (s_{1}, \dots, s_{n}) \in 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} = (u_{1}, \dots) = (s_{1}, \dots, s_{n}, max K_{(s_{1}, \dots, s_{n})}, max K_{(s_{1}, \dots, s_{n}, max K_{(s_{1}, \dots, s_{n})})}, \dots)

L_{S} = (l_{1}, \dots) = (s_{1}, \dots, s_{n}, min K_{(s_{1}, \dots, s_{n})}, min K_{(s_{1}, \dots, s_{n}, min K_{(s_{1}, \dots, s_{n})})}, \dots)

The following recursive definition holds:

u_{i} = \{\begin{matrix} s_{i}, & if i \leq n; \\ max K_{(u_{1}, \dots, u_{i - 1})}, & otherwise \end{matrix}

and

l_{i} = \{\begin{matrix} s_{i}, & if i \leq n; \\ min K_{(u_{1}, \dots, u_{i - 1})}, & otherwise \end{matrix}

It is also useful to define a notable sub sequence of

U_{S}

and

L_{S}

Definition 4.

Let

S \in 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} = ({\tilde{u}}_{1}, \dots) = (max K_{S}, max K_{(S, max K_{S})}, \dots)

{\tilde{L}}_{S} = ({\tilde{l}}_{1}, \dots) = (min K_{S}, min K_{(S, min K_{S})}, \dots)

The following recursive definition holds:

{\tilde{u}}_{i} = \{\begin{matrix} max K_{S}, & if i = 1; \\ max K_{(S, {\tilde{u}}_{1}, \dots, {\tilde{u}}_{i - 1})}, & otherwise \end{matrix}

{\tilde{l}}_{i} = \{\begin{matrix} min K_{S}, & if i = 1; \\ min K_{(S, {\tilde{l}}_{1}, \dots, {\tilde{l}}_{i - 1})}, & otherwise \end{matrix}

From theorem 1,

U_{S} \in G

and

L_{S} \in G

, while elements of

{\tilde{U}}_{S}

and

{\tilde{L}}_{S}

are all even or all odd, then

{\tilde{U}}_{S} \notin G

and

{\tilde{L}}_{S} \notin G

From definition 3, let

S = (s_{1})

, then

U_{S} = (s_{1}, s_{1} + 1, s_{1} + 3, \dots, s_{1} + 2^{n} - 1)

{\tilde{U}}_{S} = (s_{1} + 1, s_{1} + 3, \dots, s_{1} + 2^{n} - 1)

L_{S} = (s_{1}, s_{1} - 1, s_{1} - 3, \dots, s_{1} - 2^{n} + 1)

and

{\tilde{L}}_{S} = (s_{1} - 1, s_{1} - 3, \dots, s_{1} - 2^{n} + 1)

. Table 1 shows some examples of Gilbreath sequences and the closed form for

{\tilde{u}}_{n}

4. Gilbreath polynomials

Definition 5.

Let

P = (p_{1}, \dots, p_{m})

be the ordered sequence of first m prime numbers and let

P_{m} (n)

be a function such that

{\tilde{u}}_{n} = 2^{m + n - 1} + P_{m} (n)

where

P_{m} (n) = a_{m, 0} + a_{m, 1} n + \dots + a_{m, k} n^{k}

, then

P_{m}

is called m-th Gilbreath polynomial.

Through simple algebra one can prove that for the ordered sequence of first m prime numbers,

P_{m} (n)

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 $G C$

Gilbreath polynomials are closely related to prime numbers and

G C

. Let a finite sequence of integers

S = (s_{1}, \dots, s_{n})

, 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 = (p_{1}, p_{2}) = (2, 3) \in G_{2}

Gilbreath sequence formed by the first two prime numbers. From (5),

min K_{(p_{1}, p_{2})} \leq p_{2} \leq max K_{(p_{1}, p_{2})}

and from theorem 1,

(p_{1}, p_{2}, p_{2}) \in G_{3}

. By definition of P,

p_{n} > p_{n - 1}

. Since

min K_{(p_{1}, p_{2})} \leq p_{2}

, it is certainly true that

min K_{(p_{1}, p_{2})} \leq p_{3}

. The left inequality is proved for

n = 3

and it is easy to prove for every n. The proof of

min K_{(p_{1}, \dots, p_{n - 1})} \leq p_{n}

is trivial and holds for all prime numbers, hence

p_{n} ⩽ max K_{(p_{1}, \dots, p_{n - 1})} \Rightarrow G C (n)

. Given Gilbreath polynomials in definition 5,

max K_{(p_{1}, \dots, p_{n - 1})} = 2^{n - 1} + P_{n - 1} (1)

, then

p_{n} - 2^{n - 1} ⩽ P_{n - 1} (1) \Rightarrow G C (n)

(6)

Left side of (6)

p_{n} - 2^{n - 1} ⩽ P_{n - 1} (1)

(7)

consists of Gilbreath polynomial conjecture whose solution implies

G C

. Unfortunately, bounds for

p_{n}

are not enough good to prove (7) however this opens the way for a new approach to the

G C

[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

(s_{1}, \dots, s_{n}) \in G_{n}

is even or odd, then for any odd or even integer

max K_{S} \leq k \leq min K_{S}

respectively the sequence

(s_{1}, \dots, s_{n}, k) \in G_{n + 1}

Theorem 1 proves equation (6) involving Gilbreath polynomials and equation (7) implies

G C

. 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

G = a_{m, k}

and a paper on

G

will be published in the future.

References

Dusart, P. The kth Prime is Greater than k(ln k + ln ln k − 1) for k ≥ 2. Mathematics of Computation 1999, 68, 411–415. [Google Scholar] [CrossRef]
Dusart, P. Estimates of Some Functions Over Primes without R.H. 2010. [Google Scholar]
Odlyzko, A.M. Iterated Absolute Values of Differences of Consecutive Primes. Mathematics of Computation 1993, 61, 373–380. [Google Scholar] [CrossRef]
Rosser, B. The n-th Prime is Greater than n log n. Proceedings of the London Mathematical Society 1939, s2-45, 21–44. [Google Scholar] [CrossRef]
Rosser, B. Explicit Bounds for Some Functions of Prime Numbers. American Journal of Mathematics 1941, 63, 211–232. [Google Scholar] [CrossRef]
Sloane, N.J.A.; Inc., T.O.F. Sequence A347925 from the on-line encyclopedia of integer sequences, 2020.
Sloane, N.J.A.; Inc., T.O.F. Sequence A34794 from the on-line encyclopedia of integer sequences, 2020.

Table 1. Some examples of Gilbreath sequences and their closed form for

{\tilde{u}}_{n}

Table 1. Some examples of Gilbreath sequences and their closed form for

{\tilde{u}}_{n}

m	$S \in G_{m}$	${\tilde{u}}_{n}$
2	$(44, 45)$	$2^{n + 1} + 43$
3	$(21, 20, 18)$	$2^{n + 2} + 14$
4	$(38, 39, 39, 39)$	$2^{n + 3} - n^{2} - 5 n + 31$
4	$(6, 7, 5, 3)$	$2^{n + 3} - n^{2} - 3 n - 5$
5	$(28, 29, 27, 25, 21)$	$2^{n + 4} - n^{2} - 5 n + 5$
6	$(7, 8, 10, 6, 6, 6)$	$2^{n + 5} - 4 n^{2} - 20 n - 26$
6	$(13, 14, 14, 14, 12, 10)$	$2^{n + 5} - \frac{n^{4}}{12} - \frac{5 n^{3}}{6} - \frac{71 n^{2}}{12} - \frac{115 n}{6} - 22$
7	$(93, 94, 94, 94, 92, 92, 94)$	$2^{n + 6} - \frac{n^{4}}{6} - \frac{7 n^{3}}{3} - \frac{77 n^{2}}{6} - \frac{122 n}{3} + 30$

Table 2. Gilbreath polynomials for

m \leq 16

Table 2. Gilbreath polynomials for

m \leq 16

m	$P_{m} (n)$
1, 2, 3	1
4	$- n^{2} - 3 n - 1$
5	$- n^{2} - 5 n - 5$
6	$- \frac{2 n^{3}}{3} - 5 n^{2} - \frac{55 n}{3} - 19$
7	$- \frac{n^{4}}{6} - \frac{7 n^{3}}{3} - \frac{77 n^{2}}{6} - \frac{116 n}{3} - 47$
8	$- \frac{n^{5}}{30} - \frac{2 n^{4}}{3} - \frac{35 n^{3}}{6} - \frac{85 n^{2}}{3} - \frac{1277 n}{15} - 109$
9	$- \frac{n^{6}}{180} - \frac{3 n^{5}}{20} - \frac{65 n^{4}}{36} - \frac{155 n^{3}}{12} - \frac{5327 n^{2}}{90} - \frac{2579 n}{15} - 233$
10	$- \frac{n^{7}}{1260} - \frac{n^{6}}{36} - \frac{79 n^{5}}{180} - \frac{151 n^{4}}{36} - \frac{2441 n^{3}}{90} - \frac{1087 n^{2}}{9} - \frac{36481 n}{105} - 483$
11	$- \frac{n^{9}}{181440} - \frac{n^{8}}{4032} - \frac{169 n^{7}}{30240} - \frac{41 n^{6}}{480} - \frac{8389 n^{5}}{8640} - \frac{1597 n^{4}}{192} - \frac{599441 n^{3}}{11340}$
	$- \frac{1202527 n^{2}}{5040} - \frac{177197 n}{252} - 993$
12	$- \frac{n^{10}}{1814400} - \frac{13 n^{9}}{362880} - \frac{31 n^{8}}{30240} - \frac{1123 n^{7}}{60480} - \frac{20833 n^{6}}{86400} - \frac{41497 n^{5}}{17280}$
	$- \frac{3375899 n^{4}}{181440} - \frac{10094093 n^{3}}{90720} - \frac{12276223 n^{2}}{25200} - \frac{355399 n}{252} - 2011$
13	$- \frac{n^{11}}{19958400} - \frac{n^{10}}{259200} - \frac{5 n^{9}}{36288} - \frac{13 n^{8}}{4320} - \frac{27841 n^{7}}{604800} - \frac{46711 n^{6}}{86400}$
	$- \frac{46133 n^{5}}{9072} - \frac{991007 n^{4}}{25920} - \frac{50938267 n^{3}}{226800} - \frac{3525203 n^{2}}{3600} - \frac{7851061 n}{2772} - 4055$
14	$- \frac{n^{12}}{239500800} - \frac{n^{11}}{2661120} - \frac{49 n^{10}}{3110400} - \frac{299 n^{9}}{725760} - \frac{54871 n^{8}}{7257600} - \frac{5093 n^{7}}{48384}$
	$- \frac{25465669 n^{6}}{21772800} - \frac{854669 n^{5}}{80640} - \frac{60581657 n^{4}}{777600} - \frac{82179283 n^{3}}{181440} - \frac{102126421 n^{2}}{51975}$
	$- \frac{15729347 n}{2772} - 8149$
15	$- \frac{n^{13}}{3113510400} - \frac{n^{12}}{29937600} - \frac{389 n^{11}}{239500800} - \frac{269 n^{10}}{5443200} - \frac{7727 n^{9}}{7257600} - \frac{15781 n^{8}}{907200}$
	$- \frac{4919917 n^{7}}{21772800} - \frac{13119557 n^{6}}{5443200} - \frac{58181479 n^{5}}{2721600} - \frac{424785041 n^{4}}{2721600} - \frac{4521951163 n^{3}}{4989600}$
	$- \frac{3269429687 n^{2}}{831600} - \frac{292152089 n}{25740} - 16337$
16	$- \frac{n^{14}}{43589145600} - \frac{17 n^{13}}{6227020800} - \frac{73 n^{12}}{479001600} - \frac{2557 n^{11}}{479001600} - \frac{5777 n^{10}}{43545600} - \frac{4051 n^{9}}{1612800}$
	$- \frac{11564263 n^{8}}{304819200} - \frac{20564861 n^{7}}{43545600} - \frac{107393969 n^{6}}{21772800} - \frac{471325651 n^{5}}{10886400} - \frac{18807572041 n^{4}}{59875200}$
	$- \frac{2266391933 n^{3}}{1247400} - \frac{595484981809 n^{2}}{75675600} - \frac{818209547 n}{36036} - 32715$

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Submitted:

20 February 2023

Posted:

21 February 2023

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Abstract

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Subject: Computer Science and Mathematics - Algebra and Number Theory

1. Introduction to $GC$

Let the ordered sequence

P = (2, 3, 5, 7, 11, 13, 17, \dots) = (p_{1}, \dots, p_{n})

formed by prime numbers, and set

p_{a}^{b} = \{\begin{matrix} p_{a + 1} - p_{a}, & if b = 1; \\ | p_{a + 1}^{b - 1} - p_{a}^{b - 1} |, & otherwise \end{matrix}

(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

G C (n)

denote the

G C

computed on the sequence

(p_{1}, \dots, p_{n})

, then there is an interesting computational proof of

G C (n)

by A. M. Odlyzko [3] for

n < 10^{13}

2. Properties of Gilbreath sequence

Definition 1.

Let

S = (s_{1}, s_{2}, s_{3}, \dots, s_{n})

be a finite sequence of integers and

s_{a}^{b} = \{\begin{matrix} s_{a + 1} - s_{a}, & if b = 1; \\ | s_{a + 1}^{b - 1} - s_{a}^{b - 1} |, & otherwise \end{matrix}

(2)

where

b \in [1; n - 1]

, then S is called Gilbreath sequence iff

s_{1}^{b} = 1 \forall b

For example, let

S = (2, 3, 5, 9, 11, 13, 17)

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

2 3 5 9 11 13 17

1 2 4 2 2 4

1 2 2 0 2

1 0 2 2

1 2 0

1 2

Let

G_{n}

denote the set of all Gilbreath sequences of length n and

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 G_{7}

Lemma 1.

Let

S = (s_{1}, \dots, s_{n}) \in G_{n}

and

S^{'} = (s_{1}, \dots, s_{n - 1})

be finite sequences of integers, then

S^{'} \in 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 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^{'}

s_{1}^{1} = \dots = s_{1}^{n - 2} = 1

as a consequence of

S \in G_{n}

, then

S^{'} \in G_{n - 1}

. □

Definition 2.

Let

S = (s_{1}, \dots, s_{n}) \in G_{n}

and

S^{'} = (s_{1}, \dots, s_{n}, k)

be finite sequences of integers. Denote with

K_{S}

the set of integers k such that

S^{'} \in 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 G_{n}

. Gilbreath triangle of

S^{'}

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 | | \dots | | |

s_{1}^{n - 1} | s_{2}^{n - 2} - | s_{3}^{n - 3} - | s_{4}^{n - 4} - | \dots - | s_{n - 1}^{1} - | s_{n} - k | | \dots | | | |

| s_{1}^{n - 1} - | s_{2}^{n - 2} - | s_{3}^{n - 3} - | s_{4}^{n - 4} - | \dots - | s_{n - 1}^{1} - | s_{n} - k | | \dots | | | | |

where

s_{1}^{1} = \dots = s_{1}^{n - 2} = s_{1}^{n - 1} = 1

as a consequence of

S \in 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 | | \dots | | | | | = 1

, then

S^{'} \in 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 | | \dots | | | | | = 1

(3)

We will refer to equation (3) as Gilbreath equation of S. There are

2^{n}

values of k that satisfy (3), then

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 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 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 = 2 s_{n} - max K_{S}

. A remarkable relation is

max K_{S} + min K_{S} = 2 s_{n}

(5)

Lemma 2.

Let

S = (s_{1}, \dots, s_{n}) \in G_{n}

be a finite sequence of integers where

s_{1} \in 2 Z

, then

s_{2}, \dots, s_{n} \in 2 Z + 1

Proof.

Let

S_{1} = (s_{1})

, where

s_{1} \in 2 Z

. From definition 2,

S_{2} = (s_{1}, k) \in G_{2}

k = s_{1} \pm 1 \in 2 Z + 1

. Let now the sequence

S_{3} = (s_{1}, s_{1} \pm 1, k)

, from definition 2,

S_{3} \in G_{3}

k = | \pm 1 | + (s_{1} \pm 1) \pm 1 = 1 + s_{1} \pm 1 \pm 1

. From the previous step,

s_{1} \pm 1 \in 2 Z + 1

. Then

1 + s_{1} \pm 1 \in 2 Z

and

1 + s_{1} \pm 1 \pm 1 \in 2 Z + 1

. By induction, this can be proved for every element of S. If

S \in 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 = (s_{1}, \dots, s_{n}) \in G_{n}

be a finite sequence of integers where

s_{1} \in 2 Z + 1

, then

s_{2}, \dots, s_{n} \in 2 Z

Proof.

Same argument as lemma 3. □

Lemma 4.

Let

2 Z + (\frac{1}{2} \pm \frac{1}{2})

denote the sets

2 Z

and

2 Z + 1

and let a finite sequence of integers

S = (s_{1}, \dots, s_{n}) \in G_{n}

where

s_{1} \in 2 Z + (\frac{1}{2} \pm \frac{1}{2})

. Then

s_{2}, \dots, s_{n} \in 2 Z + (\frac{1}{2} \mp \frac{1}{2})

Proof.

Lemma 2 and lemma 3. □

Lemma 5.

Let

S = (s_{1}, \dots, s_{n}) \in G_{n}

and

S^{'} = (s_{1}, \dots, s_{n}, k) \in G_{n + 1}

be finite sequences of integers where

s_{1} \in 2 Z + (\frac{1}{2} \pm \frac{1}{2})

. Then

k \in K_{S} = {x \in [min K_{S}, max K_{S}] \land x \in 2 Z + (\frac{1}{2} \mp \frac{1}{2})}

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

(s_{1}, \dots, s_{n}, k) \in G_{n + 1}

where

(s_{1}, \dots, s_{n}) \in 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 = (s_{1}, \dots, s_{n}) \in G_{n}

and

S^{'} = (s_{1}, \dots, s_{n}, k)

be finite sequences of integers, then

k \in K_{S} \Leftrightarrow S^{'} \in G_{n + 1}

Proof.

Suppose that

s_{1} \in 2 Z + (\frac{1}{2} \pm \frac{1}{2})

. Prove the right implication first. From definition 2,

k \in [min K_{S}, max K_{S}]

and from lemma 5,

k \in 2 Z + (\frac{1}{2} \mp \frac{1}{2})

. Then

k \in K_{S} \Rightarrow S^{'} \in G_{n + 1}

. Prove the left implication by contradiction. Suppose that

S^{'} \in G_{n + 1}

but

k \notin K_{S}

. Then

k \in {x \notin [min K_{S}, max K_{S}] \lor x \notin 2 Z + (\frac{1}{2} \mp \frac{1}{2})}

. From definition 2 and lemma 5 it is not possible to have

S^{'} \in G_{n + 1}

k > max K_{S} \lor k < min K_{S} \lor k \notin 2 Z + (\frac{1}{2} \mp \frac{1}{2})

. Then it is true

k \in K_{S} \Leftarrow S^{'} \in G_{n + 1}

. □

Corollary 1.

Let

S = (s_{1}, \dots, s_{n})

be a finite sequence of integers, then

S \in G_{n} \Leftrightarrow min K_{(s_{1}, \dots, s_{m})} \leq s_{m + 1} \leq max K_{(s_{1}, \dots, s_{m})} \forall m \in [1, n)

Proof.

max K_{(s_{1}, \dots, s_{m - 1})}

, since

max K_{(s_{1}, \dots, s_{m - 1})}

is the maximum value for the m-th element. The same goes for

min K_{(s_{1}, \dots, s_{m - 1})}

, 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 = (s_{1}, \dots, s_{n}) \in G_{n}

be a finite sequence of integers, from (4), any solutions of the Gilbreath equation cannot be greater than

max K_{S}

, so the sequence

(s_{1}, \dots, s_{n}, max K_{S}) \in G_{n + 1}

is the upper bound sequence for S. Let now the new sequence

S^{'} = (s_{1}, \dots, s_{n}, max K_{S})

and its upper bound sequence

(s_{1}, \dots, s_{n}, max K_{S}, max K_{(s_{1}, \dots, s_{n}, max K_{S})}) \in G_{n + 2}

and so on. Equally, let a finite sequence of integers

S = (s_{1}, \dots, s_{n})

, from (4), any value of k cannot be smaller than

min K_{S}

and the new sequence

S^{'} = (s_{1}, \dots, s_{n}, min K_{S}, k)

will have the lower limit for

k = min K_{(s_{1}, \dots, s_{n}, min K_{S})}

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 = (s_{1}, \dots, s_{n}) \in 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} = (u_{1}, \dots) = (s_{1}, \dots, s_{n}, max K_{(s_{1}, \dots, s_{n})}, max K_{(s_{1}, \dots, s_{n}, max K_{(s_{1}, \dots, s_{n})})}, \dots)

L_{S} = (l_{1}, \dots) = (s_{1}, \dots, s_{n}, min K_{(s_{1}, \dots, s_{n})}, min K_{(s_{1}, \dots, s_{n}, min K_{(s_{1}, \dots, s_{n})})}, \dots)

The following recursive definition holds:

u_{i} = \{\begin{matrix} s_{i}, & if i \leq n; \\ max K_{(u_{1}, \dots, u_{i - 1})}, & otherwise \end{matrix}

and

l_{i} = \{\begin{matrix} s_{i}, & if i \leq n; \\ min K_{(u_{1}, \dots, u_{i - 1})}, & otherwise \end{matrix}

It is also useful to define a notable sub sequence of

U_{S}

and

L_{S}

Definition 4.

Let

S \in 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} = ({\tilde{u}}_{1}, \dots) = (max K_{S}, max K_{(S, max K_{S})}, \dots)

{\tilde{L}}_{S} = ({\tilde{l}}_{1}, \dots) = (min K_{S}, min K_{(S, min K_{S})}, \dots)

The following recursive definition holds:

{\tilde{u}}_{i} = \{\begin{matrix} max K_{S}, & if i = 1; \\ max K_{(S, {\tilde{u}}_{1}, \dots, {\tilde{u}}_{i - 1})}, & otherwise \end{matrix}

{\tilde{l}}_{i} = \{\begin{matrix} min K_{S}, & if i = 1; \\ min K_{(S, {\tilde{l}}_{1}, \dots, {\tilde{l}}_{i - 1})}, & otherwise \end{matrix}

From theorem 1,

U_{S} \in G

and

L_{S} \in G

, while elements of

{\tilde{U}}_{S}

and

{\tilde{L}}_{S}

are all even or all odd, then

{\tilde{U}}_{S} \notin G

and

{\tilde{L}}_{S} \notin G

From definition 3, let

S = (s_{1})

, then

U_{S} = (s_{1}, s_{1} + 1, s_{1} + 3, \dots, s_{1} + 2^{n} - 1)

{\tilde{U}}_{S} = (s_{1} + 1, s_{1} + 3, \dots, s_{1} + 2^{n} - 1)

L_{S} = (s_{1}, s_{1} - 1, s_{1} - 3, \dots, s_{1} - 2^{n} + 1)

and

{\tilde{L}}_{S} = (s_{1} - 1, s_{1} - 3, \dots, s_{1} - 2^{n} + 1)

. Table 1 shows some examples of Gilbreath sequences and the closed form for

{\tilde{u}}_{n}

4. Gilbreath polynomials

Definition 5.

Let

P = (p_{1}, \dots, p_{m})

be the ordered sequence of first m prime numbers and let

P_{m} (n)

be a function such that

{\tilde{u}}_{n} = 2^{m + n - 1} + P_{m} (n)

where

P_{m} (n) = a_{m, 0} + a_{m, 1} n + \dots + a_{m, k} n^{k}

, then

P_{m}

is called m-th Gilbreath polynomial.

Through simple algebra one can prove that for the ordered sequence of first m prime numbers,

P_{m} (n)

4.1. Relation between Gilbreath polynomials and $G C$

Gilbreath polynomials are closely related to prime numbers and

G C

. Let a finite sequence of integers

S = (s_{1}, \dots, s_{n})

, 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}

Let

P = (p_{1}, p_{2}) = (2, 3) \in G_{2}

Gilbreath sequence formed by the first two prime numbers. From (5),

min K_{(p_{1}, p_{2})} \leq p_{2} \leq max K_{(p_{1}, p_{2})}

and from theorem 1,

(p_{1}, p_{2}, p_{2}) \in G_{3}

. By definition of P,

p_{n} > p_{n - 1}

. Since

min K_{(p_{1}, p_{2})} \leq p_{2}

, it is certainly true that

min K_{(p_{1}, p_{2})} \leq p_{3}

. The left inequality is proved for

n = 3

and it is easy to prove for every n. The proof of

min K_{(p_{1}, \dots, p_{n - 1})} \leq p_{n}

is trivial and holds for all prime numbers, hence

p_{n} ⩽ max K_{(p_{1}, \dots, p_{n - 1})} \Rightarrow G C (n)

. Given Gilbreath polynomials in definition 5,

max K_{(p_{1}, \dots, p_{n - 1})} = 2^{n - 1} + P_{n - 1} (1)

, then

p_{n} - 2^{n - 1} ⩽ P_{n - 1} (1) \Rightarrow G C (n)

(6)

Left side of (6)

p_{n} - 2^{n - 1} ⩽ P_{n - 1} (1)

(7)

consists of Gilbreath polynomial conjecture whose solution implies

G C

. Unfortunately, bounds for

p_{n}

are not enough good to prove (7) however this opens the way for a new approach to the

G C

[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

(s_{1}, \dots, s_{n}) \in G_{n}

is even or odd, then for any odd or even integer

max K_{S} \leq k \leq min K_{S}

respectively the sequence

(s_{1}, \dots, s_{n}, k) \in G_{n + 1}

Theorem 1 proves equation (6) involving Gilbreath polynomials and equation (7) implies

G C

G = a_{m, k}

and a paper on

G

will be published in the future.

References

Dusart, P. The kth Prime is Greater than k(ln k + ln ln k − 1) for k ≥ 2. Mathematics of Computation 1999, 68, 411–415. [Google Scholar] [CrossRef]
Dusart, P. Estimates of Some Functions Over Primes without R.H. 2010. [Google Scholar]
Odlyzko, A.M. Iterated Absolute Values of Differences of Consecutive Primes. Mathematics of Computation 1993, 61, 373–380. [Google Scholar] [CrossRef]
Rosser, B. The n-th Prime is Greater than n log n. Proceedings of the London Mathematical Society 1939, s2-45, 21–44. [Google Scholar] [CrossRef]
Rosser, B. Explicit Bounds for Some Functions of Prime Numbers. American Journal of Mathematics 1941, 63, 211–232. [Google Scholar] [CrossRef]
Sloane, N.J.A.; Inc., T.O.F. Sequence A347925 from the on-line encyclopedia of integer sequences, 2020.
Sloane, N.J.A.; Inc., T.O.F. Sequence A34794 from the on-line encyclopedia of integer sequences, 2020.

Table 1. Some examples of Gilbreath sequences and their closed form for

{\tilde{u}}_{n}

Table 1. Some examples of Gilbreath sequences and their closed form for

{\tilde{u}}_{n}

m	$S \in G_{m}$	${\tilde{u}}_{n}$
2	$(44, 45)$	$2^{n + 1} + 43$
3	$(21, 20, 18)$	$2^{n + 2} + 14$
4	$(38, 39, 39, 39)$	$2^{n + 3} - n^{2} - 5 n + 31$
4	$(6, 7, 5, 3)$	$2^{n + 3} - n^{2} - 3 n - 5$
5	$(28, 29, 27, 25, 21)$	$2^{n + 4} - n^{2} - 5 n + 5$
6	$(7, 8, 10, 6, 6, 6)$	$2^{n + 5} - 4 n^{2} - 20 n - 26$
6	$(13, 14, 14, 14, 12, 10)$	$2^{n + 5} - \frac{n^{4}}{12} - \frac{5 n^{3}}{6} - \frac{71 n^{2}}{12} - \frac{115 n}{6} - 22$
7	$(93, 94, 94, 94, 92, 92, 94)$	$2^{n + 6} - \frac{n^{4}}{6} - \frac{7 n^{3}}{3} - \frac{77 n^{2}}{6} - \frac{122 n}{3} + 30$

Table 2. Gilbreath polynomials for

m \leq 16

Table 2. Gilbreath polynomials for

m \leq 16

m	$P_{m} (n)$
1, 2, 3	1
4	$- n^{2} - 3 n - 1$
5	$- n^{2} - 5 n - 5$
6	$- \frac{2 n^{3}}{3} - 5 n^{2} - \frac{55 n}{3} - 19$
7	$- \frac{n^{4}}{6} - \frac{7 n^{3}}{3} - \frac{77 n^{2}}{6} - \frac{116 n}{3} - 47$
8	$- \frac{n^{5}}{30} - \frac{2 n^{4}}{3} - \frac{35 n^{3}}{6} - \frac{85 n^{2}}{3} - \frac{1277 n}{15} - 109$
9	$- \frac{n^{6}}{180} - \frac{3 n^{5}}{20} - \frac{65 n^{4}}{36} - \frac{155 n^{3}}{12} - \frac{5327 n^{2}}{90} - \frac{2579 n}{15} - 233$
10	$- \frac{n^{7}}{1260} - \frac{n^{6}}{36} - \frac{79 n^{5}}{180} - \frac{151 n^{4}}{36} - \frac{2441 n^{3}}{90} - \frac{1087 n^{2}}{9} - \frac{36481 n}{105} - 483$
11	$- \frac{n^{9}}{181440} - \frac{n^{8}}{4032} - \frac{169 n^{7}}{30240} - \frac{41 n^{6}}{480} - \frac{8389 n^{5}}{8640} - \frac{1597 n^{4}}{192} - \frac{599441 n^{3}}{11340}$
	$- \frac{1202527 n^{2}}{5040} - \frac{177197 n}{252} - 993$
12	$- \frac{n^{10}}{1814400} - \frac{13 n^{9}}{362880} - \frac{31 n^{8}}{30240} - \frac{1123 n^{7}}{60480} - \frac{20833 n^{6}}{86400} - \frac{41497 n^{5}}{17280}$
	$- \frac{3375899 n^{4}}{181440} - \frac{10094093 n^{3}}{90720} - \frac{12276223 n^{2}}{25200} - \frac{355399 n}{252} - 2011$
13	$- \frac{n^{11}}{19958400} - \frac{n^{10}}{259200} - \frac{5 n^{9}}{36288} - \frac{13 n^{8}}{4320} - \frac{27841 n^{7}}{604800} - \frac{46711 n^{6}}{86400}$
	$- \frac{46133 n^{5}}{9072} - \frac{991007 n^{4}}{25920} - \frac{50938267 n^{3}}{226800} - \frac{3525203 n^{2}}{3600} - \frac{7851061 n}{2772} - 4055$
14	$- \frac{n^{12}}{239500800} - \frac{n^{11}}{2661120} - \frac{49 n^{10}}{3110400} - \frac{299 n^{9}}{725760} - \frac{54871 n^{8}}{7257600} - \frac{5093 n^{7}}{48384}$
	$- \frac{25465669 n^{6}}{21772800} - \frac{854669 n^{5}}{80640} - \frac{60581657 n^{4}}{777600} - \frac{82179283 n^{3}}{181440} - \frac{102126421 n^{2}}{51975}$
	$- \frac{15729347 n}{2772} - 8149$
15	$- \frac{n^{13}}{3113510400} - \frac{n^{12}}{29937600} - \frac{389 n^{11}}{239500800} - \frac{269 n^{10}}{5443200} - \frac{7727 n^{9}}{7257600} - \frac{15781 n^{8}}{907200}$
	$- \frac{4919917 n^{7}}{21772800} - \frac{13119557 n^{6}}{5443200} - \frac{58181479 n^{5}}{2721600} - \frac{424785041 n^{4}}{2721600} - \frac{4521951163 n^{3}}{4989600}$
	$- \frac{3269429687 n^{2}}{831600} - \frac{292152089 n}{25740} - 16337$
16	$- \frac{n^{14}}{43589145600} - \frac{17 n^{13}}{6227020800} - \frac{73 n^{12}}{479001600} - \frac{2557 n^{11}}{479001600} - \frac{5777 n^{10}}{43545600} - \frac{4051 n^{9}}{1612800}$
	$- \frac{11564263 n^{8}}{304819200} - \frac{20564861 n^{7}}{43545600} - \frac{107393969 n^{6}}{21772800} - \frac{471325651 n^{5}}{10886400} - \frac{18807572041 n^{4}}{59875200}$
	$- \frac{2266391933 n^{3}}{1247400} - \frac{595484981809 n^{2}}{75675600} - \frac{818209547 n}{36036} - 32715$

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

Abstract

1. Introduction to GC

2. Properties of Gilbreath sequence

3. Upper and lower bound sequence for P

4. Gilbreath polynomials

4.1. Relation between Gilbreath polynomials and G C

5. Conclusions and future work

References

Abstract

1. Introduction to GC

2. Properties of Gilbreath sequence

3. Upper and lower bound sequence for P

4. Gilbreath polynomials

4.1. Relation between Gilbreath polynomials and G C

5. Conclusions and future work

References

MDPI Initiatives

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1. Introduction to $GC$

4.1. Relation between Gilbreath polynomials and $G C$

1. Introduction to $GC$

4.1. Relation between Gilbreath polynomials and $G C$