Tuesday, November 15th

Problem
Find the largest positive integer $b$ such that there exists an integer a that satisfies $3\cdot2^a+1=b^2$. 


Solution
We have that $3\cdot 2^a=b^2-1=(b-1)(b+1)$, thus one of the factors $b\pm1$ has to be a power of 2. After considering the two options, we have the possible values for b to be 5 and 7. Thus $b=7$ is the largest possible value for $b$.