Integer solutions of $a^3+2a+1=2^b$ What are the solutions in integers of $a^3+2a+1=2^b$?
[Source: Serbian competition problem]

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


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*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 2b-1 is odd for all integers a and b > 0, then a must always be odd
ii.) We notice that the last digit of 2b-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)=(2b+1)(2b-1)     
