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.