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POWER CHAIN. (Posted on 2010-07-22) Difficulty: 4 of 5
{75, 100, 125} is an example of an arithmetic progression of positive integers such that the n-th term is a perfect n-th power.
Find a longer sequence with this feature.

What is the longest you can get?
P.S. Trival solution(d=0) excluded.

No Solution Yet Submitted by Ady TZIDON    
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Some Thoughts Eight is impossible Comment 6 of 6 |

It is impossible for there to be an eight term arithmetic sequence with the nth member being an nth power.  Consider the 2nd, 4th, 6th, and 8th terms.  They are all perfect squares and form an arithmetic sequence by themselves.  But it has been proven that there are no nontrivial sequences of four perfect squares.  Therefore the perfect power property of the original sequence must fail at or before the seventh power.

  Posted by Brian Smith on 2010-07-22 16:37:35
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