Some General Results and Open Questions on Palintiple Numbers

Palintiples are natural numbers which are integer multiples of their digit reversals. The most well-known base-10 example is 87912 = 4 · 21978. Using only elementary number theory we elucidate some general properties of palintiples of an arbitrary base. Palintiples naturally fall into three mutually exclusive and exhaustive classes based upon the structure of their carries. We apply our results to finding all palintiples of the class whose carries exhibit a “shifted-symmetric” structure. Moreover, we find all 5-digit palintiples whose carries are “symmetric” (the same when read forward or backward). We go on to answer some open questions posed in a paper of Pudwell while leaving several questions of our own: is divisibility of the base by the multiplier plus 1 enough to determine all symmetric palintiples? Which bases (among these is base-10) only allow for the existence of symmetric palintiples? Are there infinitely many such bases? Finally, we reveal some connections between palintiples and complex roots of unity.