Let b ≥ 2 be a numeration base. In Nițică [1], 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 [1] [2]. In [3], 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 [3] 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 [3] suggests a positive answer. This case is solved by following example.

Example 1. The palindromes 9 ∧ k and 10 ∧ k − 2 1 , k ≥ 1 are a pair of 10-wARH numbers separated of degree 2.

Let s b ( N ) denote the sum of base b digits of integer N. If x is a string of digits, let ( x ) ∧ k denote the base 10 integer obtained by repeating x k-times. Let [ x ] b denote the value of the string x in base b. If N is an integer, let N 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 ( N ) ⋅ M , respectively s b ( N ) + 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 ( [2], 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 = ( 1 + α ) l , l ≥ 0 . Assume b ≡ 2 + α ( mod 2 + 2 α ) . Define N k = [ ( 1 α ) ∧ k ] b . Then there exists M ≥ 0 integer such that

s b ( N k ) ⋅ M = ( s b ( N k ) ⋅ M ) R = N k 2 .

In particular, the numbers N k , k ≥ 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 = ( 12 ) 3 l , is also covered by ( [1], Example 10). Theorem 3 does not give any information if b = 2.

Proof. If b ≥ 3 Theorem 3 can be applied to α = b − 2 . This gives the b-wARH numbers N k = [ ( 1 α ) ∧ k ] b for k = ( 1 + α ) l , l ≥ 0 . Consider now the degrees d q = [ 1 ( b 2 − 4 b + 3 ) ∧ q ] b , q ≥ 1 .

Using that [ 1 α ] b + [ 1 ( b 2 − 4 b + 3 ) ] b = [ 1 α ] 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 [ ( 1 α ) ∧ k − q ] [ 1 ( α 1 ) ∧ q ] b form a pair of b-w ARH numbers separated of degree d q .

[ ( 1 α ) ∧ k ] b + [ 1 ( b 2 − 4 b + 3 ) ∧ q ] b = [ ( 1 α ) ∧ k − q ] [ 1 ( α 1 ) ∧ q ] b

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

[ S 10 ∧ k 1 S R ] 2 + [ 1 ∧ k 0 ∧ q ] 2 = [ S ( 1 ) ∧ k + 2 S R ] 2 .

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

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