SMARAl~ACHE CONCATENATED POWER DECIMALS

In this paper we prove that all Smarandache concatenated k-power decimals are irrational numbers. For any positive integer k, we define the Smarandache concatenated k-power decimal a k as follows: a l = 0.1234567891011..., a2 = 0.149162536496481100121... (1) a 3 = 0.18276412521634351272910001331..., ... , etc. In this peper we discuss the irrationally of ak' We prove the following result Theorem. For any positive integer k, a k is an irrational number. Proof. We nou suppose that a k is a rational number. Then, by [1, Theorem 135], a k is an infinite periodical decimal such that were r, t are fixed integers, with r~O and t>0 , aI' ... , a,., (1,.-1> ... , ar t are integers satisfYing o s; a j s; 9 (i = 1,2, ... , r-r-t). However, we see from (1) that there exist arbitrary many continuous zeros in the expansion of ak' Therefore, we find from (2) that ar-l = ... = ar-t = O. It implies that a k is a finite decimal; a contradiction. Thus, a k must be an irrational number. The theorem is proved. Finally, we pose a further question as follows: Question. Is a k a transcedental number for any positive integer k? By an old result of Mahler [2], the answer of our question is positive for k= 1. References: 1. G.H.Hardy and E.M.Wright, "An Introduction to the Theory of Numbers", Oxford University Press, Oxford, 1938. I K.Mahler, "Aritmetische Eigenschaften einer Klasse von Dezimalbruchen", NederL Akad. Wetesch. Proc., Ser.A, 40 (1937),421-428.