We determine all integers n such that n2 has at most three base-q digits for q ε (2, 3, 4, 5, 8, 16). More generally, we show that all solutions to equations of the shape where q is an odd prime, n > m > 0 and t2, πMπ,N < q, either arise from "obvious" polynomial families or satisfy m ≤ 3. Our arguments rely upon Padé approximants to the binomial function, considered q-adically.
|Title of host publication||Number Theory - Diophantine Problems, Uniform Distribution and Applications|
|Subtitle of host publication||Festschrift in Honour of Robert F. Tichy's 60th Birthday|
|Publisher||Springer International Publishing AG|
|Number of pages||26|
|Publication status||Published - 1 Jun 2017|
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