Show that any positive number of the form 4n-1 is divisible by a prime of the same form.
We know that there are primes of the form 2, or of the form
Primes of the form
don't exist (except 2 which is a prime of the form
and the
only one of this sort)
Let be any number and
let its prime factorization be
where
are all primes
If none of these primes are of the form they must either be
2 or of the form
Note that
If one of them is 2 then
for
This means
can possibility only be
But is
modulo 4
So this contradicts our assumption that none of the primes in
the prime factorization
are of the form
So that every number of the form must have a prime
factor of the same form
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