Integer solutions of $a^3+2a+1=2^b$

Question

What are the solutions in integers of $a^3+2a+1=2^b$?

[Source: Serbian competition problem]

Answer 1

My suggestion would be to start with a(a+2)=2^b-1

1. We also notice that when a is odd, a²+2 is odd. likewise if a is even, a²+2 is even
   i.) From here, we can generalize that since 2^b-1 is odd for all integers a and b > 0, then a must always be odd

   ii.) We notice that the last digit of 2^b-1 ossiclates between the numbers 1,3,7,5

   iii.)From ii.) the remaining values of the last digit of a can be reduced to 3, 5 and 7
   (i.e. b mod 4= 0 or 2 ; b is a multiple of 2)

Combining all the conditions, we can form the diophantine equation: a(a²+2)=(2^b+1)(2^b-1)