Let be the least possible for which there are and for which
, where p is a prime and . Also , , .
That there are such x,y and m is the subject of the previous exercise.
Show and unless .
By taking as a factor of both and , I find, from (1) above, that must be factored by . But p is prime. Therefore .
I feel I must have overlooked something, because I have not used the hypothesis that is minimal. What have I overlooked, please?
The above problem is from GE Andrews's "Number Theory", ex. 2, Chapter 11-2.