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Number Theory Integer solution

jacks

Well-known member
Apr 5, 2012
226
Find no. of Integer value of $\left(a,b\right)$ which satisfy $4^a+4a^2+4 = b^2$
 

Amer

Active member
Mar 1, 2012
275
Find no. of Integer value of $\left(a,b\right)$ which satisfy $4^a+4a^2+4 = b^2$
take a =2

[tex] 4^2 + 4(2)^2 + 4 = 4(4 + 4 + 1) = 4(9) [/tex]
[tex] b^2 = 36 \Rightarrow b = \pm 3 [/tex]
 

topsquark

Well-known member
MHB Math Helper
Aug 30, 2012
1,123
take a =2

[tex] 4^2 + 4(2)^2 + 4 = 4(4 + 4 + 1) = 4(9) [/tex]
[tex] b^2 = 36 \Rightarrow b = \pm 3 [/tex]
Actually [tex]b = \pm 6[/tex]

-Dan
 

Amer

Active member
Mar 1, 2012
275
try a = 4, b = 18
 
Last edited:

Opalg

MHB Oldtimer
Staff member
Feb 7, 2012
2,708
Find no. of Integer value of $\left(a,b\right)$ which satisfy $4^a+4a^2+4 = b^2$
To see that there are no solutions with $a>4$, notice first that $b$ must be even. Next, the equation $4^a+4a^2+4 = b^2$ can be written as $(2^a)^2+4a^2+4 = b^2$. Since $2^a$ is even, and $b$ is also even, the smallest possible value for $b$ would be $2^a+2$. But $(2^a+2)^2 = 4^a + 2^{a+2} + 4.$ Therefore $$b^2 = 4^a+4a^2+4 \geqslant 4^a + 2^{a+2} + 4 ,$$ from which $4a^2 \geqslant 2^{a+2}$ and hence $a^2\geqslant 2^a.$ That only happens when $a\leqslant 4.$