Find the value of 217 -1 , 218 -1 and 219 +1

 

Recreational Mathematics

Find the value of 2¹⁷ -1 , 2¹⁸ -1 and 2¹⁹ +1

When the exponent is even   2¹⁸ -1 =( 2⁹)² – 1²  = (2⁹ -1) (2⁹ +1)

2⁹ = 512; 2¹⁸ -1 = (511) (513) = (500+11) (500+13 ) = 500² + 500(11+13) + 11x13 = 250000 + 12000 + 143 = 262143

When the exponent is odd

2¹⁷ -1= 2 .2¹⁶  -2  +1 = 2 (2¹⁶ – 1 ) + 1

                                   = 2[(2⁸ -1)(2⁸ +1)] +1 = 2[(256-1)(256+1)] +1

                                                                      = 2(255)(257) +1

                                                                      = 2[(250+5)(250+7)] +1

                                                                       =2[250² +250 x 12 +35] +1

                                                                        = 2[62500 + 3000 + 35]+1

                                                                        = 2(65535) +1= 131071

2¹⁹ +1 = 2.2¹⁸ -2 + 3 =2(2¹⁸ – 1) + 3

                                   = 2[(2⁹ -1)(2⁹ +1)] +3 = 2[(512-1)(512+1)] +3

                                                                      = 2(511)(513) +3

                                                                      = 2[(500+11)(500+13)] +3

                                                                       =2[500² +500 x 24 +143] +3

                                                                        = 2[250000 +12000 +143]+3

                                                                       = 2(262143) +1=  524286 + 3= 524289