Open Journal of Discrete Mathematics
Vol.09 No.03(2019), Article ID:93592,5 pages
10.4236/ojdm.2019.93007
Infinite Sets of Solutions and Almost Solutions of the Equation
Viorel Niţică, Premalata Junius
Department of Mathematics, West Chester University of Pennsylvania, West Chester, USA
Copyright © 2019 by author(s) and Scientific Research Publishing Inc.
This work is licensed under the Creative Commons Attribution International License (CC BY 4.0).
http://creativecommons.org/licenses/by/4.0/
Received: May 23, 2019; Accepted: July 6, 2019; Published: July 9, 2019
ABSTRACT
Motivated by their intrinsic interest and by applications to the study of numeric palindromes and other sequences of integers, we discover infinite sets of solutions and almost solutions of the equation . Most of our results are valid in a general numeration base.
Keywords:
Sequences of Integers, Infinite Sets
1. Introduction
In this paper, motivated by their intrinsic interest and by applications to the study of numeric palindromes and other sequences of integers, we discover infinite sets of solutions and almost solutions of the equation
(1)
An almost solution of (1) is a pair of integers for which the equality (1) holds up to a string of digits for which we understand the position and all entries. Most of our results are valid in a general numeration base. Recently one of us showed in Nitica [1] that, in any numeration base b, for any integer N not divisible by b, the equation (1) has an infinity of solutions . Nevertheless, as one can see from [3] , finding explicit values for M can be difficult from a computational point of view, even for small values of N, e.g. . We show here explicit infinite families of solutions of (1) that are valid in all numeration bases.
Another application of our results may appear in the study of the classes of multiplicative and additive Ramanujan-Hardy numbers, recently introduced in Nitica [2] . The first class consists of all integers N for which there exists an integer M such that , the sum of base b-digits of N, times M, multiplied by the reversal of the product, is equal to N. The second class consists of all integers N for which there exists an integer M such that , times M, added to the reversal of the product, is equal to N. As showed in Nitica [1] [2] , the solutions of Equation (1) for which we can compute the sum of digits of or of , can be used to find infinite sets of above numbers.
2. Statements of the Main Results
Let be a numeration base. If x is a string of digits, let denote the base 10 integer obtained by repeating x k-times. Let denote the value of the string x in base b.
Theorem 1. a) Let a numeration base. Let , be integers. Then the product
(2)
is a palindrome.
b) Let . Then the product (2) is a palindrome if and almost a palindrome if .
The proof of Theorem 1 is done in Section 3.
The computations from Table 1 illutrate the result from Theorem 1 if and .
Theorem 2. a) Let a numeration base. Let , be integers. Then the product
(3)
is almost a palindrome.
The proof of Theorem 2 is done in Section 4.
The computations from Table 2 illustrate the result from Theorem 2 if and .
Table 1. The product (2) if and .
Table 2. The product (3) if and .
3. Proof of Theorem 1
Proof. We first assume that and distinguish three cases:
Case 1: We show that the product (2) is equal to the palindrom
Case 2: We will use repeatedly that . We show that the product (2) is equal to the palindrome
Case 3: We show that the product (2) is equal to the palindrom
Assume now that the Cases 1 and 3 follow verbatim. In Case 2 the product 2 becomes the almost palindrome
4. Proof of Theorem 2
Proof. We first assume that and distinguish three cases:
Case 1: We show that the product (3) is equal to the almost palindrome
(4)
(5)
Case 2: We show that the product (3) is equal to the almost palindrome:
Case 3: We show that the product (3) is equal to the almost palindrome:
Assume now that . The Cases 1, 2 and 3 follow verbatim. If the Cases 1 and 2 follow verbatim. In Case 3 the product 3 becomes the almost palindrome
5. Conclusion
Motivated by possible applications to the study of palindromes and other sequences of integers we find integer solutions and almost solutions of the equation . The results are valid in a genral numeration base. Our results support the following conjecture: for integers , such that , the product is a palindrome or almost a palindrome. Theorem 1 covers the case and Theorem 2 covers the case .
Conflicts of Interest
The authors declare no conflicts of interest regarding the publication of this paper.
Cite this paper
Niţică, V. and Junius, P. (2019) Infinite Sets of Solutions and Almost Solutions of the Equation
References
- 1. Niţică, V. (2019) Infinite Sets of B-Additive and B-Multiplicative Ramanujan-Hardy Numbers. Journal of Integer Sequences, 22, Article 9.4.3.
- 2. Niţică, V. (2018) About Some Relatives of the Taxicab Number. Journal of Integer Sequences, 21, Article 18.9.4.
- 3. World of Numbers. http://www.worldofnumbers.com/em36.htm