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5

Part of 1988 Austrian-Polish Competition

Problems(1)

b_k = a_k + 9 and a_{k+1} = 8b_k + 8, a_k does not have perfect squares

Source: Austrian Polish 1988 APMC

4/30/2020
Two sequences (ak)k≥0(a_k)_{k\ge 0} and (bk)k≥0(b_k)_{k\ge 0} of integers are given by bk=ak+9b_k = a_k + 9 and ak+1=8bk+8a_{k+1} = 8b_k + 8 for k≥0k\ge 0. Suppose that the number 19881988 occurs in one of these sequences. Show that the sequence (ak)(a_k) does not contain any nonzero perfect square.
Perfect Squarenumber theoryrecurrence relationSequence