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All numbers in the sequence are different - [ILL 1977]

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January 11, 2011
number theory unsolvednumber theory

Problem Statement

The sequence an,k ,k=1,2,3,…,2n ,n=0,1,2,…,a_{n,k} \ , k = 1, 2, 3,\ldots, 2^n \ , n = 0, 1, 2,\ldots, is defined by the following recurrence formula: a1=2,an,k=2an−1,k3,,an,k+2n−1=12an−1,k3a_1 = 2,\qquad a_{n,k} = 2a_{n-1,k}^3, \qquad , a_{n,k+2^{n-1}} =\frac 12 a_{n-1,k}^3\text{for}   k = 1, 2, 3,\ldots, 2^{n-1} \ , n = 0, 1, 2,\ldots Prove that the numbers an,ka_{n,k} are all different.
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