Update SHORT_NOTES.md

Author: raihankhan

SHORT_NOTES.md

@@ -22,16 +22,23 @@ if P is a prime number and a number n can be represeted by n=P^k . Then, There a
 
 
               Phi(n)=Phi(P^k)
+              
                     =n-P^(k-1)
+                    
                     =P^k - P^(k-1)
                    
 If  n = P1^a1 * P2^a2 * P3^a3 ................ Pk^ak , where P1,P2....Pk are prime numbers then , We can say that
 
       Phi(n) = Phi(P1^a1) * Phi(P2^a2) .......... Phi(Pk^ak)
+      
              = ( P1^a1 - P1^(a1-1) ) * (P2^a2 - P2^(a2-1) ) ......... ( Pk^ak - Pk^(ak-1) )
+             
              = ( P1^a1 - P1^a1/P1 ) ) * (P2^a2 - P2^a2/P2 ) ) ......... ( Pk^ak - Pk^ak/Pk) )  [a^(b-1) = a^b/a ]
+             
              = P1^a1 ( 1-1/P1 ) P2^a2 ( 1-1/P2 ) .........Pk^ak( 1-1/pk)
+             
              = P1^a1 * P2^a2 .....Pk^ak * ( 1-1/P1 )( 1-1/P2 ).........( 1-1/pk)
+             
              = n * ( 1-1/P1 )( 1-1/P2 ).........( 1-1/pk)