A Special Class of Reverse Digit Multiples

She thinks it’s all pointless, jejune artifice,

But everything rhymes in the nation of spice.

As discussed in another note, for a given base B and constant integer k there may be integers N such that kN = R_B(N), where R_B(N) signifies the number with reversed digits of N in base B. For example, with B=10 and k=4 we have 4*2178 = 8712.

For an n-digit number we can denote the digits of N (in the base B) from least to most significant as d₀, d₁, d₂, …, d_n, and we can let c₀, c₁, c₂, … c_n-1 denote the sequence of carries when multiplying N by k. We stipulate the c_-1 = c_n = 0, and for convenience we define the column vectors for matched pairs of digits and carries as follows

Then, in matrix form, the conditions on successive pairs of digits and carries from the outside inwards, as given in the previous note, can be written as

where

Since the components of C (i.e., the values of the carries) are always in the range from 0 to k-1, we can show the values of the components of D as two k x k arrays, and noting the viable solutions give integer values. For certain combinations of B and K we get a remarkably complete and orderly set of solutions.  An example is with B=29, k=5. In this case, noting that the boundary carries are [c_-1, c_n]^T = [0,0], the viable pairs of digits are those on the left hand column in the table below.

The first numbers in each pair are incremented by columns, and the second numbers are incremented by rows.  This makes it symmetrical, in the sense that every pair of digits (a,b) is symmetrical with a pair of digits (b,a).  The digits 5, 11, 17, 23 do not appear in any solution. A two digit solution is [1][6] in base 29, meaning that 5(1*29+6) = (1+6*29). The values of 1 and 6 are along the diagonal of the k x k arrays showing the components of the D vectors, so the carries in this pair are equal, [1,1]^T, and the next pair of digits needs to come from that menu. To expand to a 4-digit solution, we could select, say, the digits [9][19], giving the solution [1][9][19][6]. Since these two new digits imply the next carries [3,3]^T, we would need to select a pair of digits from the [3,3]^T menu.  Each menu has a set of repeated digits, i.e., [0][0], or [1][1], or [2][2], and so on. We could insert just one of these to make a solution with an odd number of digits. But let’s select [22][27] for the next pair of digits, so we now have constructed the 6-digit example [1][9][22][27][19][6], and since this pair came from the [4,4]^T carries, the next pair would need to come from that menu. Suppose we select [24][4], which corresponds to the [0,0]T carries, so we could choose the pair of digits [0][0] from that menu, and since it corresponds again to zero carries, we would choose [3][18] for the next pair of digits. If we stop at this point, we have a 12-digit solution

and we can directly verify that

with B=29.  One of the interesting things about this special class of solutions is that the base polynomial is divisible by a polynomial whose coefficients are the carries. Thus the denominator in the fraction above factors as

The second factor has the matched coefficients 1, 3, 4, 0, 0, 3, 0, 0, 4, 3, 1, which is the sequence of carries when multiplying N by 5.

We have examples of this “complete” structure when B = k² + k – 1. Thus with k = 2 or 3 we have the examples

Young (1992) and Sloane (2014) wrote nice papers on reverse digit multiples, and describes the sets of solutions in terms of graphs. Each node of the graph corresponds to a pair of carries, and the digit pairs are shown as transitions from one node to another. To illustrate, the figure below gives a depiction of the case B=11, k=3.

There are also similar structures that  involve increments larger than 1, as shown be the following examples.

In these special cases the solutions are all along the diagonal of the k x k arrays, so we can identify each carry index by the respective row, but in the general case, the menu items would include the relevant carry pair along with each digit pair on the menus. This directs us to the next menu when we select a digit pair from the current menu.

This subject is interesting because of how the reversal requirement leads to the basic elements being pairs of digits working inwards from opposite ends of the number. This implies distant correlations, even though there is a large amount of freedom in the sequence of digits. Indeed we could randomly select from each “menu” at each stage of the process, so the values themselves are unpredictable but the correlations are necessarily maintained to arrive at any self-consistent string. The string is extended, not at either end, but from the middle.

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