On digit sums of multiples of an integer |
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Authors: | Cécile Dartyge Pantelimon St?nic? |
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Institution: | a Institut Élie Cartan, Université Henri Poincaré-Nancy 1, BP 239, 54506 Vandoeuvre Cedex, France b Instituto de Matemáticas, Universidad Nacional Autónoma de México, C.P. 58089, Morelia, Michoacán, Mexico c Department of Applied Mathematics, Naval Postgraduate School, Monterey, CA 93943-5216, USA |
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Abstract: | Let g>1 be an integer and sg(m) be the sum of digits in base g of the positive integer m. In this paper, we study the positive integers n such that sg(n) and sg(kn) satisfy certain relations for a fixed, or arbitrary positive integer k. In the first part of the paper, we prove that if n is not a power of g, then there exists a nontrivial multiple of n say kn such that sg(n)=sg(kn). In the second part of the paper, we show that for any K>0 the set of the integers n satisfying sg(n)?Ksg(kn) for all k∈N is of asymptotic density 0. This gives an affirmative answer to a question of W.M. Schmidt. |
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Keywords: | 11N25 11N37 |
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