( ) * DIRECT Proof *

(21<+35-1 =(411+121<+9 ) - 1

th > I & OPD

LlAwI 8112Mt 1) 2-1 for tfm 70

BASI CAST : m=o,

(2.0+1)=1= 1-1=0 → 810

Induction step : IF 8/121<472-1 THEN 8/(212+3) 2- /

=Assume 8/121<+42-1 ⇒ 121<+1)2- I = 8L,IEZ

8l= ( 21<+1 ) 2- / = (415+4)<+1) - 1 Hook+ 8) - ( 81<+8 )=

.

=[( 4k '

+121<+9 ) -1 ] - ( ooktp )8l - [ (212+372-1) - 8( 1<+1 ) Rkt 3) 2-1=81+8 CKH )

( Pl ) ) a P (2) a .. . n PCK ) ) → PCKH )

= = =

BASEC_AsE NI ! 2 is PRIME :

Induction STEP IF PCZ ) n PC 3) n - - . PCK ) → Pckti )

⇒I CASEI : KH is prvme⇒ DONE

Caste KH ( s composite⇒ KH - A. b 21 a ,b< ktl

By Itt : A. b CAN BE written AS a product OF PRIME

⇒ KH =p.gg;bgyq .

.= promoof PRIME

'

EE

. .

= = =

=

BASIE :

Fo =o EVEN

F. = 1 < rap

=

= =

BASE CASE : n= 12 = 6+6-

n= 13 = 4+7

n= 14 = 4+4+6

h= 15 = 9th

÷suction Aypottts FOR 12 El ± K WE can

MAFEEXALHYK nvggits ( To prove: KH nuggets )

12+1=(12-3)+4-

K - 3112 ⇒ make

T 4 5TH.k→nggasBVYTHESE Boy A 4- PIECE .

many nrggets