Numbers with anticlosed sets of non-squares

[jens at voss-ahrensburg.de]

Zitat von hv at crypt.org:

> jens at voss-ahrensburg.de wrote:
> :Let R be a ring. We will call a subset T of R *anticlosed* if for any
> :elements t1 and t2 of T, neither the sum t1 + t2 nor the product t1 * t2
> :lies in T.
> :
> :For an integer n >= 2, let R := Z/nZ, and let T be the set of non-squares
> :of R. For which n is this set T anticlosed?
> :
> :I ran a quick search with the computer and found that within the first
> :10000 integers, only 2, 3, 5, 6, 7 and 10 have this property, so it
> :appears that this definition may yield a finite (and quite short)
> :sequence. Any idea on how to prove that no numbers beyond 10 have an
> :anticlosed set of non-squares?
>
> Am I misunderstanding something or is '7' a typo for '9'?

No, the '7' is the effect of a bug in my program ;(

>  n = 7 => T = { 3, 5, 6 } and 3 + 3 = 6

Correct, the 7 should be out!

>  n = 9 => T = { 2, 5, 8 }, and [ T, + ] = [ T, * ] = { 1, 4, 7 }

If I am not mistaken here, T = {2, 3, 5, 6, 8} and 2 + 3 = 5, so 9
is out, too.

As a note: It is not too hard to show that no number divisible by 4
can appear in the list.