Keywords:Expansion in a base, palindromes, antipalindromes, palindromic numbers, antipalindromic numbers.
Everybody has certainly heard about palindromes: words that stay the same when read backwards. For instance, kayak, radar, or rotor. Mathematicians are interested in palindromic numbers: positive integers whose expansion in a certain integer base is a palindrome. The following problems are studied: palindromic primes, palindromic squares and higher powers, multi-base palindromic numbers, etc. In this paper, we define and study antipalindromic numbers: positive integers whose expansion in a certain integer base is an antipalindrome. We present new results concerning divisibility and antipalindromic primes, antipalindromic squares and higher powers, and multi-base antipalindromic numbers. We provide a user-friendly application for all studied questions.
Comment after publishing (November 2021)
A reader has found numbers 43490045690, 45741687530, 51300998210, 56554588825 and 667298874550 that are antipalindromic in bases 6 and 8. Therefore 6 and 8 are not suitable candidates in the open problem (4.).
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Copyright (c) 2021 Ľubomíra Dvořáková, Stanislav Kruml, David Ryzák
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