Infinite Sets of Related b-wARH Pairs

Abstract

Let b ≥ 2 be a numeration base. A b-weak additive Ramanujan-Hardy (or b-wARH) number N is a non-negative integer for which there exists at least one non-negative integer A, such that the sum of A and the sum of base b digits of N, added to the reversal of the sum, give N. We say that a pair of such numbers are related of degrees d ≥ 0 if their difference is d. We show for all numeration bases an infinity of degrees d for which there exists an infinity of pairs of b-wARH numbers related of degree d.

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Nitica, C. and Nitica, V. (2020) Infinite Sets of Related b-wARH Pairs. Open Journal of Discrete Mathematics, 10, 1-3. doi: 10.4236/ojdm.2020.101001.

1. Introduction

Let b ≥ 2 be a numeration base. In Nițică , motivated by some properties of the taxicab number, 1729, we introduced the class of b-additive Ramanujan-Hardy (or b-ARH) numbers. It consists of non-negative integers N for which there exist at least an integer M ≥ 1 such that the product of M and the sum of base b digits of N, added to the reversal of the product, give N. Many examples of b-ARH numbers can be found in  . In , we introduced the class of b-weak-additive Ramanujan-Hardy (or b-wARH) numbers. It consists of non-negative integers N for which there exist at least an integer A ≥ 0, such that the sum of A and the sum of base b digits of N, added to the reversal of the sum, give N. It is shown in  that the class of b-wARH numbers contains the class of b-ARH numbers. Moreover, the class of b-wARH numbers contains all numerical palindromes with an even number of digits or with an odd number of digits and the middle digit even.

We say that a pair of b-wARH numbers are related of degree d ≥ 0 if their difference is d. Our main result shows, for all numeration base b ≥ 2 an infinity of degrees d for which there exists an infinity of pairs of b-wARH numbers related of degree d. Our main result leaves open the case when b = 10 and d = 2, which is of strong particular interest and for which Table 1 in  suggests a positive answer. This case is solved by following example.

Example 1. The palindromes ${9}^{\wedge k}$ and ${10}^{\wedge k-2}1,k\ge 1$ are a pair of 10-wARH numbers separated of degree 2.

2. The Statement of the Main Result

Let ${s}_{b}\left(N\right)$ denote the sum of base b digits of integer N. If x is a string of digits, let ${\left(x\right)}^{\wedge k}$ denote the base 10 integer obtained by repeating x k-times. Let ${\left[x\right]}_{b}$ denote the value of the string x in base b. If N is an integer, let ${N}^{\mathcal{R}}$ denote the reversal of N, that is, the number obtained from N writing its digits in reverse order. The operation of taking the reversal is dependent on the base. In the definition of a b-ARH number or a b-wARH number N we take the reversal of the base b representation of ${s}_{b}\left(N\right)\cdot M$, respectively ${s}_{b}\left(N\right)+A$. The following Theorem is our main result.

Theorem 2. For all numeration bases b ≥ 2 there exists an infinity of degrees d ≥ 0 for which there exists an infinity of pairs of b-wARH numbers related of degree d.

Theorem 2 is proved in Section 3. The following Theorem is ( , Theorem 1) and it is a crucial ingredient in the proof of our main result, Theorem 2.

Theorem 3. Let α ≥ 1 integer, b ≥ α + 1 integer, and $k={\left(1+\alpha \right)}^{l},l\ge 0$. Assume $b\equiv 2+\alpha \left(\mathrm{mod}2+2\alpha \right)$. Define ${N}_{k}={\left[{\left(1\alpha \right)}^{\wedge k}\right]}_{b}$. Then there exists M ≥ 0 integer such that

${s}_{b}\left({N}_{k}\right)\cdot M={\left({s}_{b}\left({N}_{k}\right)\cdot M\right)}^{R}=\frac{{N}_{k}}{2}$.

In particular, the numbers ${N}_{k},k\ge 1$, are b-ARH numbers and consequently also b-wARH numbers.

Remark 4. The particular case b = 10, α = 2, of Theorem 2, which gives ${N}_{k}={\left(12\right)}^{{3}^{l}}$, is also covered by ( , Example 10). Theorem 3 does not give any information if b = 2.

3. Proof of Theorem 2

Proof. If b ≥ 3 Theorem 3 can be applied to $\alpha =b-2$. This gives the b-wARH numbers ${N}_{k}={\left[{\left(1\alpha \right)}^{\wedge k}\right]}_{b}$ for $k={\left(1+\alpha \right)}^{l},l\ge 0$. Consider now the degrees ${d}_{q}={\left[1{\left({b}^{2}-4b+3\right)}^{\wedge q}\right]}_{b},q\ge 1$.

Using that ${\left[1\alpha \right]}_{b}+{\left[1\left({b}^{2}-4b+3\right)\right]}_{b}={\left[1\alpha \right]}_{b}$, the following computation, in which the right hand side is a palindrome with an even number of digits, shows that the numbers ${N}_{k}$ and $\left[{\left(1\alpha \right)}^{\wedge k-q}\right]{\left[1{\left(\alpha 1\right)}^{\wedge q}\right]}_{b}$ form a pair of b-w ARH numbers separated of degree ${d}_{q}$.

${\left[{\left(1\alpha \right)}^{\wedge k}\right]}_{b}+{\left[1{\left({b}^{2}-4b+3\right)}^{\wedge q}\right]}_{b}=\left[{\left(1\alpha \right)}^{\wedge k-q}\right]{\left[1{\left(\alpha 1\right)}^{\wedge q}\right]}_{b}$

Assuming $k\ge q$, this finishes the proof of the theorem if b ≥ 3. Assume now b = 2. Consider the degrees ${d}_{k,q}={\left[{1}^{\wedge k}{0}^{\wedge q}\right]}_{2},k\ge \text{1},q\ge \text{1}$. Let S be a string of length q with 0 and 1 digits. The following computation shows that the palindromes ${\left[S{10}^{\wedge k}1{S}^{R}\right]}_{2}$ and ${\left[S{\left(1\right)}^{\wedge k+2}{S}^{R}\right]}_{2}$ form a pair of 2-wARH numbers separated of degree ${d}_{k,q}$.

${\left[S{10}^{\wedge k}1{S}^{R}\right]}_{2}+{\left[{1}^{\wedge k}{0}^{\wedge q}\right]}_{2}={\left[S{\left(1\right)}^{\wedge k+2}{S}^{R}\right]}_{2}$.

Conflicts of Interest

The authors declare no conflicts of interest regarding the publication of this paper.

  Nițică, V. (2018) About Some Relatives of the Taxicab Number. Journal of Integer Sequences, 21, Article 18.9.4.  Nițică, V. (2019) Infinite Sets of b-Additive and b-Multiplicative Ramanujan-Hardy Numbers. Journal of Integer Sequences, 22, Article 19.4.3.  Nițică, V. and Török, A. About Some Relatives of Palindromes. arXiv:1908.00713.  Nițică, V. High Degree b-Niven Numbers, to Appear in Integers. http://arxiv.org/abs/1807.02573     customer@scirp.org +86 18163351462(WhatsApp) 1655362766  Paper Publishing WeChat 