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An Investigation Into the Order of Integral Powers of Set of All Natural Numbers
Published On August 10, 2024
Journal Issue LJRS Volume 24 Issue 10

An Investigation Into the Order of Integral Powers of Set of All Natural Numbers

Dr. Ladan Umaru Ibrahim
Dr. Ladan Umaru Ibrahim
¶ ∐
An Investigation Into the Order of Integral Powers of Set of All Natural Numbers
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Research ID 02ZT8

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− Abstract

This article established a fact on the order of difference of integral powers of all sets of natural numbers. The analysis was proof by use of established property of difference operator and principle of mathematical induction. The result proved conclusively that “if the elements of an arithmetic progression of set of natural numbers with positive common difference are raised to positive power k, then the kth difference is equal to the product of the common difference raised to power k (dK) and k factorial (k!).

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I. INTRODUCTION

A set of positive integers is a set of natural numbers. A set is a collection of well-defined elements. A set of natural number is a sequence, defined with respect to a constant value called a common difference (d). The terms of a sequence are defined with respect to three parameters, the first term (a), the numbers of terms in the sequence (n) and the common difference (d). The sequence of natural numbers expressed mathematically as:

\[T_{n} = a + (n - 1) d \dots\dots I\]
\[For all a\geq 1, d>1, n\in (0,\infty)\]

Any sequence generated with the above formular is called an arithmetic progression.

A perfect power is a rational root chase. An integral perfect power is the irrational root that is an integer. A finite difference is a mathematical expression of the form, and a forward difference is of the form

This article investigates all Arithmetic Progression of set of natural numbers with positive common difference. Further investigation reveals that there is a significant relationship between the difference of powers of set of natural numbers and factorial, which is represented symbolically as:

\[\Delta^{k} \left[ a + (n - 1) d \right]^{k} = d^{k} K! \tag{II}\]
\[\mathrm{Forall} a \geq 1, d \geq 1, k > 1 \mathrm{and}\]

For some

The implication of equation (II) is on the coefficient of the kth factorial, which sows the value of k is the power of the common difference and at the same time is the power of the sequence under consideration for which the difference of the set is equal to the product of the common difference raised to power k and .

An examination of any set of natural numbers having common difference greater than 2 e.g. {2, 5, 8, 11, 14, 17, 20, 23, 26, 29} reveals that the pattern is sustained (See Table I and Appendix A).

The set of perfect squares of the set is . The actual set of the first difference is . Furthermore, the set of the second difference is , a singleton set. Clearly, the set of differences of the above second difference set is . The emerging pattern motives the investigation of whether the pattern will persist for all set of natural numbers with positive common difference d > 2. The set of the second difference of the set of perfect squares clearly shows that the element of the singleton set .

\[This shows that for d = 3, k = 2.\]
\[\Delta^{2} \left[ a + (n - 1) d \right] ^ {2} = d^{k}. K!, where\]
\[a = 2, d = 3, k = 2, \mathrm{and} n \in (0, 9)\]
\[\Rightarrow \Delta^{2} [a + (n - 1)d]^{2} = 18 = 3^{2}(2!)\]

In their contribution in Exponential Diophantine Equations, Shorey and Tijdeman investigated perfect powers at integral values of a polynomial with rational integer coefficients and obtain in particular the following. Let be a polynomial with rational coefficients and with at least two simple rational zeros.

Suppose , , x and y with /Y/ > 1 are rational integers. Then the equation implies that m is bounded by a computable number depending only on b and f. Also in their contribution Ladan, Tanko, Aliyu, Ahmad and Kabiru , they investigated integral powers of polynomials with Binomial coefficients. The result of their investigation shows that "the disposition of powers of polynomials with binomial coefficients generates even positive factorial (2k)!

Also in a research work by Ladan, Aliyu, Tanko, Ahmad and Kabiru , on the location of points on the plane and the order of disposition of sum of powers of cardinal coordinates. The result of their work proved conclusively that "the sum of the powers of cardinal points is equal to the coefficients of the Binomial expansion with respect to the Pascal triangle pattern and entries". In their contribution in the difference of perfect powers of integers, Ladan, Ukwu and Apine investigated order of integral perfect powers and proved that "if any number of consecutive integers are raised to a positive integral power k, then the difference is equal to k! Based on the generalization of the theorem in this article, it shows that Ladan, Ukwu and Apine work has common difference d = 1, for all k > 1.

Similarly, in a research conducted by Ladan, Emmanuel and Tanko , investigated order of product of perfect powers and proved conclusively that if any number of consecutive integer are raised to a positive power k, then the difference of the product of the power of two consecutive integers is equal to . This established a relation between difference of powers of natural numbers and factorial. For more definitions, see [8, 9, 10.....].

There was no discussion on the relationship between the difference of powers of set of all natural numbers and factorial with respect to the common difference which is pertinent to this article. Review of literature shows that no such investigation has been undertaken. Thus, this article adds to the existing body of knowledge.

II. METHODS

2.1 Preliminary Definitions

In what follows, the difference of finite order will be defined.

2.1.1 Differences of Order One (1)

Given a sequence , defined the difference of order one at with respect to the sequence by:

\[\Delta(f_j) = f_{j+1} - f_j, \text{for every} j\]

2.1.2 Higher Order Differences

Higher order differences can be defined recursively by:

\[\Delta^{k}(f_{j}) = \Delta(\Delta^{k-1}(f_{j})) = \Delta^{k-1}(\Delta(f_{j})) = \Delta^{k-1}[f_{j+1} - f_{j}] \mathrm{for} \geq 2\]
\[\text { III. RESULTS AND DISCUSSION }\]

3.1 Preliminary Theorem

Suppose for all j belong to set of natural numbers.

Then:

(i) is natural number (ii)

(iii)

Proof

Let

Then Table I below yields value of for selected values of k and p in the set and respectively, when

Table 1: Difference Order Table for Selected Elements of Set of Natural Numbers

$f_j$ $f_j^2$ $\Delta(f_j^2)$ $\Delta^2(f_j^2)$ $\Delta^3(f_j^2)$
2421180
52539180
86457180
1112175180
1419693180
17289111180
20400129180
23529147180
2667616518
29841

It is clear that Where d = 3, k = 2, and j ∈ {2, 5,... 26, 29}.

That is

Obviously, the theorem is valid for as observed from column 4. The proofs of (i), (ii) and (iii) are direct.

(i) , which is natural number for all , proving (i).

(ii)

The principle of mathematical induction is needed in the proof of (iii).

For ,

Consequently

For all , proving , for all positive integer . This established the proof of (iii).

3.2 Main Theorem

Let

Let be a sequence defined by , for all , and .

Let , for .

We shall consider the case for the power of to establish the proof of the theorem.

\[Case (1), k = 1, n = {1, 2, 3,...}\]

are constant.

is the sequence.

The first difference is given as:

\[\begin{array}{r l} \Delta^{1}[j_{n}]^{1} & = \Delta^{1}[a + (n - 1)d]^{1}, \\\Delta[j_{n}] & = \Delta(a + dn - d) \\& = \Delta a + \Delta dn - \Delta d \\& = \Delta dn \\& = d[(n + 1) - n] \\& = d(1) \\& = d \\& = d^{1}(1!) \end{array}\]
\[Therefore, for k = 1\]
\[\Delta^ {1} [ \mathrm {j_ {n}} ] ^ {1} = \Delta^ {1} [ a + (n - 1) d ] ^ {1} = d ^ {1} (1!)\]
\[Case (2), k = 2, n = {1, 2, 3,...}\]

is set of perfect squares.

\[\Delta^ {2} [ \mathrm{j_ {n}} ] ^ {2} = \Delta^ {2} [ a + (n - 1) d ] ^ {2},\]
\[\begin{array}{r l} & {\Delta [ \Delta \mathrm{j} _ {\mathrm{n}} ^ {2} ] = \Delta^ {2} [ a + d n - d ]} \\& {\qquad = \Delta^ {2} [ (a + d n - d) (a + d n - d) ]} \\& {\qquad = \Delta^ {2} [ a + a d n - a d + a d n + d ^ {2} n ^ {2} - d ^ {2} n - a d - d ^ {2} n + d ^ {2} ]} \\& {\qquad = \Delta^ {2} [ a ^ {2} + 2 a d n - 2 a d + d ^ {2} n ^ {2} - 2 d ^ {2} n + d ^ {2} ]} \end{array}\]
\[\begin{array}{r l} & {= \Delta [ \Delta (a ^ {2} + 2 a d n - 2 a d + d ^ {2} n ^ {2} - 2 d ^ {2} n + d ^ {2}) ]} \\& {= \Delta [ \Delta a ^ {2} + 2 a d \Delta n - \Delta 2 a d + d ^ {2} \Delta (n ^ {2}) - 2 d ^ {2} \Delta (n) - \Delta d ^ {2} ]} \\& {= \Delta [ 2 a d ((n + 1) - (n) + d ^ {2} ((n + 1) ^ {2} - (n) ^ {2}) - 2 d ^ {2} ((n + 1) - (n) ]} \\& {= \Delta [ 2 a d (1) - d ^ {2} ((n ^ {2} + 2 n - 1) - (n) ^ {2} - 2 d ^ {2} (1) ]} \\& {= \Delta [ 2 a d + d ^ {2} (2 n - 1) - 2 d ^ {2})} \\& {= \Delta 2 a d + \Delta 2 d ^ {2} n - \Delta 3 d ^ {2}} \\& {= \Delta 2 d ^ {2} n} \\& {= 2 d ^ {2} \Delta (n)} \\& {= 2 d ^ {2} (n + 1 - n)} \\& {= 2 d ^ {2} (1) = \mathrm{d} ^ {2} 2!} \end{array}\]

Therefore,

\[\Delta^{2}[\mathrm{j}_{n}]^{2} = \Delta^{2}[a + (n - 1)d]^{2} = d^{2}(2!), \text{satisfied}.\]

Case (3), ,

\[\begin{array} { r l } & { \Delta ^ { 3 } [ j _ { n } ] ^ { 3 } = \Delta ^ { 3 } [ a + ( n - 1 ) d ] ^ { 3 } } \\ & { = \Delta ^ { 3 } [ ( a + d n - d ) ^ { 3 } ] } \\ & { = \Delta ^ { 3 } [ ( a + d n - d ) ( a + d n - d ) ^ { 2 } ] } \\ & { = \Delta ^ { 3 } [ ( a + d n - d ) ( a ^ { 2 } + 2 a d n - 2 a d - 2 d ^ { 2 } n + d ^ { 2 } n ^ { 2 } + d ^ { 2 } ) ] } \\ & { = \Delta ^ { 3 } [ a ^ { 3 } + 2 a ^ { 2 } d n - 2 a ^ { 2 } d - 2 a d ^ { 2 } n + a d ^ { 2 } n ^ { 2 } + a d ^ { 2 } + a ^ { 2 } d n + 2 a d ^ { 2 } n ^ { 2 } - 2 a d ^ { 2 } n ^ { 2 } - } \\ & { 2 d ^ { 3 } n ^ { 2 } + d ^ { 3 } n ^ { 3 } + d ^ { 3 } n - a ^ { 2 } d - 2 a d ^ { 2 } n + 2 a d ^ { 2 } + 2 d ^ { 3 } n - d ^ { 3 } n ^ { 2 } - d ^ { 3 } ] } \\ & { = \Delta ^ { 3 } [ a ^ { 3 } + 3 a ^ { 2 } d n - 3 a ^ { 2 } d - 6 a d ^ { 2 } n + 3 a d ^ { 2 } n ^ { 2 } + 3 a d ^ { 2 } - 3 d ^ { 3 } n ^ { 2 } + d ^ { 3 } n ^ { 3 } + 3 d ^ { 3 } n - d ^ { 3 } ] } \\ & { = \Delta ^ { 2 } [ \Delta ( a ^ { 3 } + 3 a ^ { 2 } d n - 3 a ^ { 2 } d - 6 a d ^ { 2 } n + 3 a d ^ { 2 } n ^ { 2 } + 3 a d ^ { 2 } - 3 d ^ { 3 } n ^ { 2 } + d ^ { 3 } n ^ { 3 } + 3 d ^ { 4 } n - } \\ & { d ^ { 3 } ) ] } \\ & { = \Delta ^ { 2 } [ \Delta ( a ^ { 3 } + \Delta 3 a ^ { 2 } d n - \Delta 3 a ^ { 2 } d - \Delta 6 a d ^ { 2 } n + \Delta a d ^ { 2 } n ^ { 2 } + \Delta 3 a d ^ { 2 } - \Delta 3 d ^ { 3 } n ^ { 2 } + \Delta d ^ { 4 } n ^ { 3 } +} \\ & { \Delta 3 d ^ { 3 } n - \Delta d ^ { 3 } ) ] } \\ & { = \Delta ^ { 2 } [ 3 a ^ { 2 } d \Delta n - 6 a d ^ { 2 } \Delta n + a d ^ { 2 } \Delta n ^ { 2 } + 3 d ^ { 3 } \Delta n ^ { 2 } + d ^ { 3 } \Delta n ^ { 3 } + 3 d ^ { 4 } \Delta n ] } \\ & { = \Delta ^ { 2 } [ 3 a ^ { 2 } d ( n + 1 - n ) - 6 a d ^ { 2 } ( n + 1 - n ) + a d ^ { 2 } ( ( n + 1 ) ^ { 2 } - n ^ { 2 } ) - 3 d ^ { 3 } ( ( n + 1 ) ^ { 2 } -} \\ & { n ^ { 2 } ) + d ^ { 3 } ( ( n + 1 ) ^ { 3 } - n ^ { 3 } ) + 3 d ^ { 4 } ( n + 1 - n ) ] } \\ & { = \Delta ^ { 2 } [ 3 a ^ { 2 } d - 6 a d ^ { 2 } ( n ^ { 2 } + 2 n + 1 - n ^ { 2 } ) - 3 d ^ { 3 } n ^ { 2 } + 2 n + 1 - n ^ { 2 } + d ^ { 3 } ( [ n + 1 ) ( n ^ { 2 } +} \\ & { \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \textnormal{or} . } \\ & { = \Delta^ { 2 } [ 3 a ^ { 2 } d - 6 a d ^ { 2 } + a d ^ { 2 } ( 2 n + 1 ) - 3 d ^ { 3 } ( 2 n + 1 ) + d ^ { 3 } ( n ^ { 3 } + 3 n ^ { 2 } + 3 n + 1 ) - n ^ { 3 } +} \\ & = \Delta^ { 2 } [ a b e c t ] \end{array}\]
\[= \Delta^{2}[3a^{2}d - 5ad^{2} + 2ad^{2}n - 3d^{3}n - 2d^{3} + 3d^{3}n^{2} + 3d^{3}]\]

Therefore,

\[\Delta^{3} [\mathrm{j_{n}}]^{3} = \Delta^{3} [a + (n - 1) d]^{3} = d^{3}(3!)\]

Thus, the proof of the theorem is established for the values integral power of . This clearly reveals that the pattern is sustained for the values of k belong to natural numbers for all d > 2 and .

Assume the validity of the theorem of for some natural number .

Inductively, the theorem holds for k = 1, see case (1)

The truth of theorem for k = 1, implies is true for all positive k > 1, that is , for some k > 1. (See Case 2 and 3).

Thus by induction hypothesis, the truth of the theorem for k implies the validity of the theorem for .

\[\Delta^ {k+1} [ j_ {n} ] ^ {k+1} = \Delta^ {k+1} [ a + (n - 1) d ] ^ {k + 1}\]
\[\Delta^{k+1} [ a + (n - 1) d ]^{k+1} = \mathsf{d}^{k+1} (k + 1)!\]

This completes the proof, that is the pattern is sustained for all powers of set of natural numbers, for all and , , , .

The theorem is established.

3.3 Corollary

for every arithmetic progression of set of natural numbers and for any positive integer k, for all . In other words, for any positive integer k, the order difference of the power of a sequence of natural numbers is equal to where d is the common difference, .

By implication, the theorem states that "if the elements of an arithmetic progression of the set of natural numbers with positive common difference are raised to a positive power k, then the difference is equal to the product of the common difference and k factorial .

3.4 Remarks

The following strong relationship exist between the difference operator and D operator (differential operator).

\[D ^ {k} (K ^ {p}) = 0 \text { if } k > p\]
\[D ^ {k} x ^ {k} = k!\]

The coefficient of

\[D^{k}(K^{p}) = \frac{P!}{(p-k)} x^{p-k} \text{for} k \geq 2, k < p \text{is even.}\]

IV. CONCLUSION

This article established the structures of finite difference orders with respect to the powers of set of natural numbers. The result reveals a startling relationship between the common difference of the sequence and k!, as reflected in the theorem.

Competing Interests: Authors have declared that no competing interest exist.

− Conflict of Interest

The authors declare no conflict of interest.

− Ethical Approval

Not applicable

− Data Availability

The datasets used in this study are openly available at [repository link] and the source code is available on GitHub at [GitHub link].

− Funding

This work did not receive any external funding.

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    LCC Code: QA241
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  • Issue date

    10 August 2024

  • Language

    en

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arithmetic progression. Finite difference integral order mathematical induction positive powers
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