User:Idk12345678
My Solutions
Some Proofs I wrote
if
is prime.
Proof: Expanding out, all the coefficients are of the form
by the binomial theorem. To prove the original result we must show that if
and
, then
. Because
,
, which is divisible by
, so the original expression must be divisible by
. However if
is prime,
, since
does not contain
(because
). Therefore, in order for
to be divisible by
,
is divisible by
. All the coefficients of the expansion(besides the coefficients of
and
) are of the form
, and
, so they cancel out and
if
is prime.