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 RB(hx+1)=hRB(x)+1, where RB reverses base-B digits, for every B≥3, 2≤ 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≤ 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.
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BibTeX
@misc{panraksa20261729,
title = {{1729 in the Mirror: Reversal and Affine Multiplication}},
author = {Chatchawan Panraksa},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.22290},
eprint = {2609.22290},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.22290},
}
RIS
TY - UNPB
TI - 1729 in the Mirror: Reversal and Affine Multiplication
AU - Chatchawan Panraksa
PY - 2026
DA - 2026/09/13
PB - arXiv
DO - 10.48550/arxiv.2609.22290
UR - https://arxiv.org/abs/2609.22290
AB - 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≥3, 2≤ 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≤ 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.
ER -