Power Coding Theorem and its Ramifications

Peter Wills and I discovered empirically in 2002 that for any integer power n in any prime base b representation, the set of integers with l coding digits followed by any fixed non zero digit r, are recoded (as permutations or bijective mappings) in b-1 ways in their nth powers. The l digit coding region within the powers is offset with a shift from the position of the original l digits, and a formula for this shift was derived. The bijective (often pseudorandom) mapping on the l penultimate digits, means that every combination of l digits base b appears once and once only in the coding regions of this set of powers. After many years of attending to other matters, we completed the proof (now available as an online reprint on arXiv) in 2019 and generalised the result to all powers of numbers. The result was also extended to include an unrestricted recoding result including the case r=0 above. 

Actually, these mappings although often appearing random can sometimes be rather beautiful as for the case n=2 in base 2 

See BioMIP Web Pages for my research group 2004-2017 :  John McCaskill 2019