Abstract. We determine all integers n such that n 2 has at most three base-q digits for q ∈ {2, 3, 4, 5, 8, 16}. More generally, we show that all solutions to equations of the shapewhere q is an odd prime, n > m > 0 and t 2 , |M |, 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.
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