Question

Show that every odd prime can be put either in the form $4k+1$ or $4k+3(i.e.,4k-1),$ where k is a positive integer.

Answer 1

Let n be any odd prime. If we divide any n by 4, we get
$n=4k+r$
where $0\le r\le 4$ i.e., $r=0,1,2,3$
$\therefore eithern=4korn=4k+1$
or $n=4k+2orn=4k+3$
Clearly, 4n is never prime and
$4n+2=2(2n+1)$ cannot be prime unless $n=0$
(since, 4 and 2 cannot be factors of an odd prime).
$\therefore$ An odd prime n is either of the form
$4k+1or4k+3$
But $4k+3=4(k+1)-4+3=4k^{\prime }-1$
(where $k^{\prime }=k+1$)
$\therefore$ An odd prime n is either of the form
$4k+1or(4k+3)i.e.,4k^{\prime }-1$