A073608

[Dean Hickerson]

Jim Nastos wrote:

> Nowhere is there an implication that 1 is a prime power. If the author
> assumed that, then the sequence would be 1,2,3,4,5,6.

Since 1 *is* a prime power, the definition of the sequence should explicitly
state that it's being excluded.  It's also not necessary to say "is a prime
or a prime power", since every prime is a prime power.

Don Reble wrote:

>     There is an elementary proof that no set of seven integers of that
>     kind exists.

Jim asked:

> Okay. Any elaboration or reference on that?

If the set has 7 integers, then 4 of them have the same parity.  Call them
a, b, c, and d, with a<b<c<d.  All of their differences are even, hence
powers of 2.  Since b-a, c-b, and their sum, c-a, are all powers of 2,
b-a and c-b must be equal.  Similarly c-b and d-c are equal.  But then
d-a = 3(b-a) is not a power of 2.

Dean Hickerson
dean at math.ucdavis.edu