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 OJDM  Vol.10 No.3 , July 2020
Infinite Sets of Solutions and Almost Solutions of the Equation N⋅M = reversal(N⋅M) II
Abstract: Motivated by their intrinsic interest and by applications to the study of numeric palindromes and other sequences of integers, we discover a method for producing infinite sets of solutions and almost solutions of the equation N⋅M = reversal (N⋅M), our results are valid in a general numeration base b > 2.

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 a method for producing infinite sets of solutions and almost solutions of the equation:

N M = r e v e r s a l ( N M ) . (1)

where if N is an integer written in base b, which is understood from the context then reversal(N) is the base b integer obtained from N writing its digits in reverse order.

An almost solution of (1) is a pair of integers ( M , N ) for which the equality (1) holds up to a few of digits for which we understand their position. Our results are valid in a general numeration base b > 2 and complement the results in [1]. Recently one of us showed in Nitica [2] that, in any numeration base b, for any integer N not divisible by b, the Equation (1) has an infinite set of solutions ( N , M ) . 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. N = 81 . We show in [1] for many numeration bases explicit infinite families of solutions of (1). This families of solutions here complement and are independent of those shown in [1].

Another application of our results may appear in the study of the classes of b-multiplicative and b-additive Ramanujan-Hardy numbers, recently introduced in Nitica [4]. The first class consists of all integers N for which there exists an integer M such that S b ( N ) , 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 S b ( N ) , times M, added to the reversal of the product, is equal to N. As showed in Nitica [2] [4], the solutions of Equation (1) for which we can compute the sum of digits of S b ( N ) M + r e v e r s a l ( S b ( N ) M ) or of S b ( N ) M r e v e r s a l ( S b ( N ) M ) , can be used to find infinite sets of above numbers.

2. Statements of the Main Results

The heuristics behind our results is that the product of a palindrome by a small integer still preserves some of the symmetric structure of the palindrome; if, in addition, the palindrome has many digits of 9, many times the results observed in base 10 can be carried over to an arbitrary numeration base b replacing 9 by b − 1.

Let b 2 be a numeration base. If x is a string of digits, let ( x ) ^ k denote the base b integer obtained by repeating x k-times. Let [ x ] b denote the value of the string x in base b.

Next theorem is one of our main results.

Theorem 1. Let b 2 be a numeration base. Let 0 < A , B , c , d b integers such that A B = [ c d ] b and c + d = A . Then,

A ^ k B = [ c A ^ k 1 d ] b .

Proof of Theorem 1 is covered in Section 3. Similar proof to that of Theorem 1 gives also the somewhat stronger statement Theorem 3.

The above table illustrates the result from Theorem 1 if b = 10 and ( A , B ) = ( 9 , 9 ) , [ c d ] b = [ 81 ] 10 , and k { 2 , 3 , 4 , 5 , 6 , 7 , 8 } . Note that 9 × 9 = 81 and 8 + 1 = 9 .

Theorem 2. Let b > 2 numeration base and k , l > 1 integers then one has:

( b 1 ) ^ k [ a 1 a 2 a 3 a l ] b = [ a 1 a 2 a 3 a l ] b [ a 1 a 2 a 3 a l 1 ] b ( b 1 ) ^ ( k l ) [ b l a 1 a 2 a 3 a l ] b (2)

in particular if b is odd and [ a 1 a 2 a 3 a l ] b = ( b l 1 ) / 2 .

Then (2) gives a solution of (1).

The proof of Theorem 2 is done in Section 4.

The following examples illustrate the statement of Theorem 2.

Example:

9 ^ 130 [ 123 ] 10 = [ 122 9 ^ 1327 83 ] 10

7 ^ 130 [ 123 ] 8 = [ 1227 ^ 127 489 ] 8

9 ^ 130 [ 123 ] 10 = [ 122 9 ^ 127 389 ] 8

Theorem 3. let b > 2 umeration base. Let 0 < A , B , c , d , α b integers such that A B = [ c d ] b and c + d = α . Then,

A ^ k B = [ c α ^ k 1 d ] b = A B ^ k

Next theorem shows for all numeration bases examples of pairs ( A , B ) that satisfy the hypothesis of Theorem 1.

Theorem 4. Let b 2 be a numeration base. Then the pairs ( A B ) = [ ( b 1 ) ( b k ) ] b , 1 k b satisfy the hypothesis of Theorem 1.

Proof:

[ ( b 1 ) ( b k ) ] b

b 2 b k b + k = b ( b k 1 ) + k = [ [ b k 1 ] , k ] b

b k 1 + k = b 1.

Corollary. Let b 2 be numeration base. Then [ ( b 1 ) ( b 2 ) ] b .

Consequently, satisfies the hypothesis of Theorem 1, consequently

( b 1 ) ^ k ( b 2 ) = [ ( b 3 ) ( b 1 ) ^ ( k 1 ) 2 ] b .

Proof: apply Theorem 4 to the pair ( A B ) = ( b 1 ) ( b 2 ) .

The above table illustrates the result from Theorem 1 & Theorem 3 if b = 7 , b 1 = 6 , b 2 = 5 , [ c d ] b = [ 42 ] 7 , thus A = 6 , B = 5 and k { 2 , 3 , 4 , 5 , 6 , 7 , 8 } . Note that [ 6 5 ] 7 = [ 42 ] 7 and [ 4 + 2 ] 7 = 6 .

| b ( A , B ) 2 3 ( 2 , 2 ) 4 ( 2 , 3 ) , ( 3 , 2 ) , ( 3 , 3 ) 5 ( 2 , 3 ) , ( 2 , 4 ) , ( 3 , 2 ) , ( 3 , 4 ) , ( 4 , 2 ) , ( 4 , 3 ) , ( 4 , 4 ) 6 ( 2 , 5 ) , ( 3 , 5 ) , ( 4 , 5 ) , ( 5 , 2 ) , ( 5 , 3 ) , ( 5 , 4 ) , ( 5 , 5 ) 7 ( 2 , 4 ) , ( 2 , 6 ) , ( 3 , 3 ) , ( 3 , 5 ) , ( 3 , 6 ) , ( 4 , 2 ) , ( 4 , 4 ) , ( 4 , 6 ) , ( 5 , 3 ) , ( 5 , 6 ) , ( 6 , 2 ) , ( 6 , 3 ) , ( 6 , 4 ) , ( 6 , 5 ) , ( 6 , 6 ) 8 ( 3 , 7 ) , ( 4 , 7 ) , ( 5 , 7 ) , ( 6 , 7 ) , ( 7 , 2 ) , ( 7 , 3 ) , ( 7 , 4 ) , ( 7 , 5 ) , ( 7 , 6 ) , ( 7 , 7 ) 9 ( 2 , 5 ) , ( 2 , 8 ) , ( 3 , 4 ) , ( 3 , 8 ) , ( 4 , 3 ) , ( 4 , 5 ) , ( 4 , 6 ) , ( 4 , 7 ) , ( 4 , 8 ) , ( 5 , 2 ) , ( 5 , 4 ) , ( 5 , 6 ) , ( 5 , 8 ) , ( 6 , 4 ) , ( 6 , 5 ) , ( 6 , 8 ) , ( 7 , 4 ) , ( 7 , 8 ) , ( 8 , 2 ) , ( 8 , 3 ) , ( 8 , 4 ) , ( 8 , 5 ) , ( 8 , 6 ) , ( 8 , 7 ) , ( 8 , 8 ) 10 ( 2 , 9 ) , ( 3 , 4 ) , ( 3 , 7 ) , ( 3 , 9 ) , ( 4 , 6 ) , ( 4 , 9 ) , ( 5 , 9 ) , ( 6 , 4 ) , ( 6 , 7 ) , ( 6 , 9 ) , ( 7 , 3 ) , ( 7 , 6 ) , ( 7 , 9 ) , ( 8 , 9 ) , ( 9 , 2 ) , ( 9 , 3 ) , ( 9 , 4 ) , ( 9 , 5 ) , ( 9 , 6 ) , ( 9 , 7 ) , ( 9 , 8 ) , ( 9 , 9 ) |

The above table shows all pairs ( A , B ) that satisfy the hypothesis of Theorem 1 for small numeration bases. We observe that for b = 2 there are no pairs ( A , B ) that satisfy the hypothesis of Theorem 1.

3. Proof of Theorem 1

l = 1 k A b l B = l = 1 k A B b l = l = 1 k ( c b + d ) b l = l = 1 k c b l + 1 + d l = 1 k b l = c b k + 1 + l = 1 k 1 c b + l = 1 k 1 d b + d b k = c b k + 1 + l = 1 k 1 ( c + d ) b l + d b k = c b k + 1 + l = 1 k 1 A b + d b k = [ c ( A ) ^ k 1 d ] b

4. Proof of Theorem 2

Using that ( b 1 ) k = b k 1 and that ( b 1 ) k l = b k l 1 .

One has that:

( b 1 ) k [ a 1 a 2 a 3 a l ] b = ( b k 1 ) [ a 1 a 2 a 3 a l ] b = [ + b k a 1 a 2 a 3 a l ] b b l [ a 1 a 2 a 3 a l ] b = + [ + b k a 1 a 2 a 3 a l ] b 1 + b k + b l b l = + [ + b k a 1 a 2 a 3 a l ] b 1 + b l ( b k l 1 ) + [ b l a 1 a 2 a 3 a l ] b = 1 ( b 1 ) ^ ( k l ) [ b l a 1 a 2 a 3 a l ] b

5. Conclusion

Motivated by possible applications to the study of palindromes and other sequences of integers we discover a method for producing infinite families of integer solutions and almost solutions of the equation N M = r e v e r s a l ( N M ) . Our results complement the results in [1] and are valid in all numeration bases b > 2 .

Acknowledgements

While working on this project C. E. was an undergraduate student at West Chester University of Pennsylvania.

Cite this paper: Nitica, V. and Ekinci, C. (2020) Infinite Sets of Solutions and Almost Solutions of the Equation N&#8901;M = reversal(N&#8901;M) II. Open Journal of Discrete Mathematics, 10, 69-73. doi: 10.4236/ojdm.2020.103007.
References

[1]   Nitica, V. and Junius, P. (2019) Infinite Sets of Solutions and Almost Solutions of the Equation NM = reversal (NM). Open Journal of Discrete Math, 9, 63-67.
https://doi.org/10.4236/ojdm.2019.93007

[2]   Nitica, V. (2019) Infinite Sets of b-Additive and b-Multiplicative Ramanujan-Hardy Numbers. The Journal of Integer Sequences, 22, Article number: 9.4.3.

[3]   World of Numbers.
http://www.worldofnumbers.com/em36.htm

[4]   Nitica, V. (2018) About Some Relatives of the Taxicab Number. The Journal of Integer Sequences, 21, Article number: 18.9.4.

 
 
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