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.

Thus,

(1)

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.