Extending Relationships (3)

This process of self-similarity in a complementary fashion, whereby a number when multiplied by another whole number is equal to its reverse is very rare indeed with respect to our customary denary system.

In fact there are only two types of situation where it occurs.

The first is the somewhat trivial case where the whole number multiple = 1. This in fact entails that the number is a palindrome, which could be alternatively stated by saying that a zero result results from subtracting the number from its reverse.

So when a number is a palindrome e.g. 343, when this number is multiplied by 1, the reverse number is obtained (which is identical to the starting number). So 343 * 1 = 343.

The other case, which again is somewhat trivial, results when a number - starting in 0 and with other digits forming a palindrome, is multiplied by 10 (i.e. the base number).

Here when the staring number is multiplied by 10, the reverse number is obtained.

So  for example 099 * 10 = 990.

However, if we exclude the validity of the starting digits (or digits) of the original number being equal to 0, then no examples of this latter form of behaviour exist in our customary number base.

Therefore, bearing this exclusion in mind,  the only case in base 10, where starting number * k = reverse (where k is a whole number) is where k = 1 (such that the number is a palindrome).

However such similarity is much more prevalent in other number bases.


Now, we have already seen that in number bases, 2, 5, 8, ... that the most striking form of such self-similarity arises where the whole number multiple (k) = 2. This implies that

reverse – original (starting) number = original number.

This coincides with the octagonal sequence

1, 8, 21, 40, ... (with k^th term =  3n² – 2n, where k = 1, 2, 3, ...).

However in base 2, this could be deemed as a somewhat trivial result that should be excluded.

We have already dealt with this issue in base 10 whereby a number (starting in one or more 0's), when multiplied by 10 can result in the reverse number being generated.

This can equally be associated with any number base. Therefore in base 2 when in the simplest case, we multiply 01 (the 1st number of the sequence) by 2 we obtain 10 (i.e. the reverse).

However if we exclude original numbers that start with one or more zeros in base 10, then we should equally exclude then in base 2.
 
Therefore, from this perspective, the first non-trivial case arises in base 5, whereby we have already seen (in the simple 2-digit case) that 8 (in denary terms) = 13 (in base 5), so that  13 * 2 = 31 (with 31 – 13 = 13)  .

Then in base 8, 21 (in denary terms) = 25 (in base 8) so that 25 * 2 = 52 (with 52 – 25 = 25) and in base 11, 40 (in denary terms) = 37 (in base 11) so that 37 * 2  = 73 (i.e. 73 – 37 = 37) and so on.

So in all these bases (2, 5, 8, 11, ...) just one example of such self similarity exists.

This equally applies then by extension to 3-digit, 4-digit, 5-digit, ...    numbers with just one valid example in each case.

Thus in base 5, in the 3-digit case, 143 * 2 = 341 (so that 341 – 143 = 143). This number is obtained by inserting the number that is 1 less than the base number in question (i.e. 5) between first and last digits.

Then again in base 5, in the 4-digit case, 1443 * 2 = 3441 (so that 3441 – 1443 = 1443). So here an additional 4 is placed anywhere between first and last digits.

And to illustrate further for the 5-digit case, 14443 * 2 = 34441 (so that 34441 – 14443 = 14443) with again an additional 4 placed between first and last digits.


Then when the whole number multiple, k = 3 so that original (starting) number * 3 = reverse, we found that this will exist in number bases 3, 7, 11, 15, ...  

This is associated with the number sequence  

2, 12, 30, 56, ...  (with k^th term =  4n² – 2n, where k = 1, 2, 3, ...).

Now again (as always the case), the initial base leads to a somewhat trivial result, where for example in the 2-digit case, where we express the 1st term of the sequence in base 3, 02 * 3 = 20.

Thus, excluding this case, the first non-trivial example occurs in base 7, where 12 (in denary terms) = 15 (in base 7) and 51 = 15 * 3.  

Then in base 11, 30 (in denary terms) = 28 (in base 11) and 82 = 28 * 3.

And in base 15, 56 in denary terms) = 3B (in base 15) and B3 = 3B * 3.

So in each of these bases, again just one 2-digit case arises with respect to this form of self-similarity.

However once again, the 2-digit-case can be extended to 3-digit, 4-digit, 5-digit, ... numbers by the continual inclusion between the first and last digits of the digit that is 1 less than the number base in question.

Thus the sole 3-digit example in base 7 of this form of self-similarity, (where reverse = original number * 3) is 165 so that 561 = 165 * 3.

And the sole 4-digit example in base 7 is 1665 so that 5661 = 1665 * 3.

And finally to illustrate the sole 5-digit example is 16665 so that 56661 = 16665 * 3.