Without direct evaluation, determine whether 102¹⁹⁹¹ + 103¹⁹⁹¹ is a proper power of an integer.

102¹⁹⁹¹ + 103¹⁹⁹¹ = 205*(a mess)

205 = 5*41

Last digits of small odd values of a mess for 102^(2n-1) + 103^(2n-1)(even ones are fractional):

{1,7,5,3,9,9,3,5,7,1,1..}

5 divides a mess iff 5 divides (2n-1)

5 does not divide 1991.

Therefore 102¹⁹⁹¹ + 103¹⁹⁹¹  is not a proper power of an integer.

Edited on May 24, 2015, 4:10 am

Posted by broll on 2015-05-24 03:49:02