Every positive integer can be written as the sum of three palindromes, numbers that remain the same when their digits are reverse. For example, 389 = 11 + 55 + 323. This holds not just for base 10 but for any base *b* ≥ 5.

[Update: a new paper on arXiv extends this to *b* = 3 and 4 as well. Base 2 requires four palindromes.]

The result and algorithms for finding the palindromes was published online last August and is in the most recent print issue of Mathematics of Computation.

Javier Cilleruelo, Florian Luca and Lewis Baxter. Every positive integer is a sum of three palindromes. DOI: https://doi.org/10.1090/mcom/3221

That Hat-trick Theorem plus Osaka Univ. Prof. Takayuki Hibi’s Palindromic-Numbers Theorem enable the decompositions of any n-D polytopes into one to three sub-polytopes whose volume(s) can be computed, and therefore, also one branch of my Esst (= equations-systems solution theories).

The below site’s Japanese Hibi-page (but not the English Hibi-page) has the photos of Hibi’s English and Japanese books; his Blue Backs pocket-book “Takakkei to Tamentai = Polygons and Polytopes” (Tokyo:

Kodansha, 2021) explains his above Theorem:

http://math.ist.osaka-u.ac.jp/course/course_hibitakayuki/