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The integral values of [log.sub.[k.sup.n]] S([n.sup.k]).(Brief article)
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Abstract: Let k, n be distinct positive integers. In this paper, we prove that [log.sub.[k.sup.n]] S(nk) is never a positive integer.
Key words: Smarandache function, logarithm, integral value.
For any positive integer a, let S(a) denote the Smarandache function of a. In [2, Problem 22], Muller posed the following problem:
Problem: Is it possible to find two distinct positive integers k and n such that [log.sub.[k.sup.n]] S([n.sup.k]) is a positive integer?
In this paper, we completely solve the above problem as follows:
Theorem: ...
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