MHB Exploring Integer Ordered Pairs in a Unique Equation

[juantheron]

The no. of positive Integer ordered pair $(a,b)$ in $4^a+4a^2+4 = b^2$

 

[kaliprasad]

The no. of positive Integer ordered pair $(a,b)$ in $4^a+4a^2+4 = b^2$

Spoiler

it may have a solution as 4^a = 2^ 2a

if

(4^a + 4a^2 + 4) >= (2^a+1)^2

or 2 * 2^a + 1 <= 4a^2 + 4

this gives a <= 6 and we need to check for a = 1 to 6.
rest is trivial

 

[juantheron]

To kaliprasad

I did not understand it

it may have a solution as $4^a = 2^ {2a}$

if $\left(4^a + 4a^2 + 4 \right) >= (2^a+1)^2$

or $2 \cdot 2^a + 1 \leq 4a^2 + 4$

this gives $a\leq 6$ and we need to check for $a =1$ to $a = 6$
.
rest is trivial

would you like to explain me.

Thanks

 

[kaliprasad]

To kaliprasad

I did not understand it

it may have a solution as $4^a = 2^ {2a}$

if $\left(4^a + 4a^2 + 4 \right) >= (2^a+1)^2$

or $2 \cdot 2^a + 1 \leq 4a^2 + 4$

this gives $a\leq 6$ and we need to check for $a =1$ to $a = 6$
.
rest is trivial

would you like to explain me.

Thanks

you know 4^a = (2^2a) = (2^a)^2

so square root is 2^a

next square has to be (2^a+ 1)^2

if 4^a + 4a^2 + 4 < (2^a+1)^2 we cannot have a pefect squre because we are less than the next square

 

[kaliprasad]

A slight different solution

Spoiler

posted at Fun with maths: Q13/056) Find Integer value of (a,b) which satisfy 4^a+4a^2+4=b^2