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__Arithmetic Progressions in Power Sequences__

For a non-zero real number *c*, consider the infinite sequence

*
*
S(c)={1^{c}, 2^{c}, 3^{c} . . . } .

*What is the cardinality **l(c)* of the longest arithmetic progression, contained in *S(c)* ?

For instance, there are *3*-terms arithmetic progressions, composed of perfect squares (like *1, 25, 49*). On the other hand, Fermat proved that no four squares form an arithmetic progression; therefore, *l(2)=3*.

A two-minutes meditation shows that if *c* is the reciprocal of a positive integer (that is, *c=1/n* for a positive integer *n*), then *S(c)* contains an *infinite* arithmetic progression. A five-minutes meditation shows then that if *l(c)* is finite, then *l(-c)* is also finite and *l(-c)=l(c)*; furthermore, if *c* is a reciprocal of a *negative integer*, then *S(c)* contains an arithmetic progression of any preassigned length. If *S(c)* contains no infinite progression and no progression of any preassigned length, then *l(c)* is finite.

It is a less trivial (but still not too complicated) task to prove that if *c* is not the reciprocal of a positive integer, then *S(c)* does not contain an infinite progression. This, however, leaves the following question open: do there exist any *c*, except for *c=±1/n* (with a positive integer *n*), such that *S(c)* contains progressions of any preassigned length?

Conjecture 1. If *c* is not a reciprocal of an integer number, then *l(c)* is finite.

Using a very strong result due to
Darmon and Merel,
I can actually
determine *l(c)* for any rational *c*. Specifically, if *c=p/q*, where *p* and *q* are positive co-prime integers, then

- if
*p=1*, then *l(c)* is infinite;
- if
*p=2*, then *l(c)=3*;
- if
*p>2*, then *l(c)=2*.

Of course, this establishes the conjecture above for all *rational* *c*. At the same time, for *c* *irrational* I can say virtually nothing about the value of *l(c)*. I would not be surprised if the following is true.
Conjecture 2. If *c* is irrational, then *S(c)* can possibly contain at most one triple of elements in an arithmetic progression. In particular, *l(c)* is at most *3* for any irrational *c*.