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Problem of the Week 931

Powerful Arithmetic Progressions

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An arithmetic progression is a sequence of integers such that the difference between successive terms is a constant d. Here are examples of nonconstant arithmetic progression of positive integers such that the kth term is a perfect kth power:

{1}
{1, 4}
{23, 25, 27}

Find a longer example.

Source: Ken D. Boklan (Baltimore, MD), American Mathematical Monthly, Problem 956, 1998. Solution in December 2000 issue. There they ask for the longest such sequence and a proof of optimality.

© Copyright 2001 Stan Wagon. Reproduced with permission.

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8 Mar 2001