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Theorem on the Distribution of the Figures 0, 1 and 2 in the Various Orders of the base 3 Representations of kth Powers of Integers Roberto Vacca - February 9, 2003 Let us consider a generic natural number n in its base 3 representation n = (h, n) 3h , with (h, n) = 0 or 1 or 2 0 The base 3 representation of the kth power of n, with k a positive integer, is n = k (h, n) 3h k , with k (h, n) = 0 or 1 or 2 0 We observe first that k (h, n) = k (h, n +3h+1) or, in other words, that k (h, n) is periodic with period function of n , due to the fact that n m mod 3h+1 implies that 3h+1 as a nk mk mod 3h+1 k (h, n) depends only on the residue of n mod 3h+1 We shall denote by Nkj(h) the number of k (h, n) which are equal to j In fact within a period (j = 0, 1, 2), that is 3h+1-1 Nkj(h) = k (h, i) i=0 The values of the ratio Nkj(h)/ 3h+1 (which are obviously equal to 1/3 for k =1 (for any h) and also for h = 0 and for k odd) can be listed as follows - denoting by Nkj(h) the ratio Nkj(h)/ 3h+1 : 1 For k = 2 with h even N20(h) = 1/3 N21(h) = (1/3) + with h odd N20(h) = (1/3) + N21(h) = N22(h) 1/3(h/2)+1 ; N22(h) = (1/3) - 1/3(h/2)+1 2/3(s+2) = (1/3) - 1/3(s+2) where s = [h/2] where s = [h/2] denoting with [x] the integer part of x. For k even and a divisor of h Nk0(h) = 1/3 Nk1(h) = (1/3) + 1/3(s+1) ; Nk2(h) = (1/3) - 1/3(s+1) where s = [h/k] For k even and not a divisor of h If k is a multiple of 3z (with z = 1, 2, 3, ...), with h <= z mod(nk) Nk0(h) = 1/3 + 2/3(s+1) where s = [h/k] (s+1) Nk1(h) = Nk2(h) = (1/3) - 1/3 where s = [h/k] For k odd and a divisor of h Nk0(h) = Nk1(h) = Nk2(h) = 1/3 For k odd and not a divisor of h If k is a multiple of 3 z with h <= z mod (nk) Nk0(h) = (1/3) + 1/3(s+1) Nk1(h) = (1/3) - 1/3(s+1) Nk2(h) = 1/3 where s = [h/k] where s = [h/k] For k even or odd and not a multiple of 3 z or a multiple of 3 z with h > z mod (nk) Nk0(h) = 1/3 + 2/3(s+2) where s = [h/k] (s+2) Nk1(h) = Nk2(h) = (1/3) - 1/3 where s = [h/k] * * * Note - A similar theorem for base 2 was published in: Gross, W.; Vacca, R. Distribution of the figures 0 and 1 in the various orders of binary representations of nth powers of integers. Math. Comp. 22 1968 423--427. 2