1729 in the Mirror: Reversal and Affine Multiplication

Abstract

The decimal reversals 1729 and 9271 have multiplicative orders of $2$ equal to 36 and 63. Motivated by this symmetry, we classify the affine commutation equation $R_B(hx+1)=hR_B(x)+1$, where $R_B$ reverses base-$B$ digits, for every $B\ge3$, $2\le h<B$ and positive integer $x$ such that neither $x$ nor $hx+1$ ends in zero. Comparing the two carry sequences separates the solutions by their digit lengths. Equal lengths force an odd number of digits alternating between two explicit intervals. When the output gains a digit, coprimality of $h$ and $B+1$ forces a repdigit input with palindromic output. In the noncoprime case, all inputs instead have even length and form a finite family. A slack-variable bijection gives the sharp maximum length, binomial counts at each length, and totals expressed through Fibonacci numbers or powers of two. For reversal pairs of primes already known to share an order index $2\le h<B$, the classification gives an exact digit criterion for reversal of their orders. Finally, an exhaustive computation with independently checkable primality certificates proves that $\{1729,9271\}$ is the unique pair of distinct odd decimal reversals whose orders of $2$ are distinct two-digit reversals of each other.