If and are in , let be the least positive remainder obtained by dividing the (ordinary) sum of and by ,
and similarly, let be the least positive remainder obtained by dividing the (ordinary) product of and by
Prove that if is a field, then either the result of repeatedly adding to itself is always different from ,
or else the first time that it is equal to occurs when the number of summands is a prime.
I look at where is the number of times 1 is added to itself.
, is prime since is a field.
when is a multiple of , .
For , .
I am really bad at this so any suggestion is very welcome!