How many positive integers 'n' are there such that [(2)^n + 1] is divisible by 7 ?

(In reply to

la meme Janette mais autrement coifee! by luminita)

An unusual approach, but I don't think its valid. For the first term, k*2^3 is even regardless of k; so if k is odd, both the first and second term are even, resulting in an even value for the right side. So it is possible to have an even value for the right side (of your second equation). What you need to show is the right side cannot be a multiple of 2, a harder task.