Squares with three nonzero digits

2016/10/31 by Bennett, Michael A., Scheerer, Adrian-Maria
#11D61 (primary) 11A63 #11J25 (secondary) #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.1610.09830

Abstract

We determine all integers n such that n² 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 shape Y² = t² + M ⋅ q^m + N ⋅ qⁿ, where q is an odd prime, n > m > 0 and t², |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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