27 Jul
Arithmetic progression devoid of powers
Find an integral arithmetic progrssion with an arbitrary large number of terms such that no term is a perfect rth power for r = 2, 3, ..., n.
Trigg gives two solutions. One is trivial, with the first term a non-power and the difference 0. The second solution is by Azriel Rosenfeld. It is quite simple, a one-liner in fact, but by no means trivial.
References
- C. W. Trigg, Mathematical Quickies, Dover, 1985, #64.
Solution
Observe that a power of an odd integer is odd, whereas a rth power (with r > 1) of an even integer is divisible by at least 4. It follows that the sequence
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