TITLE: Does there exist $a\in\mathbb N$, $b\in\mathbb Z$ that $2^na+b$ is a square for all $1\le n\le5$?
QUESTION [5 upvotes]: We consider such $a\in\mathbb N$, $b\in\mathbb Z$, o numbers of the form $2^na+b$ is square to the largest possible number of values of $n=1,2,3,4,\ldots$. 
It is easy to see that for $a = 60 $, $ b = -119 $ produced four squares: $2a+b=1=1^2$, $4a+b=121=11^2$, $8a+b=361=19^2$, $16a+b=841=29^2$. The next value $ 32a + b = 1801 $ square is no longer.
And now the actual questions:
a) Does there exist $a\in\mathbb N$, $b\in\mathbb Z$ that $2^na+b$  is a square for all $1\le n\le5$?
b) Does there exist $a\in\mathbb N$, $b\in\mathbb N$ that $2^na+b$  is a square for all $1\le n\le4$?

REPLY [2 votes]: $$a = 4110224238120;
b = 74019764836081 $$

the numbers for the item b)