Integer solutions: 3*2^x + 1 = y^2

[Math Tutor]

How many integer solutions does the equation have?
[tex]3*2^x + 1 = y^2[/tex]

[Guest]

Let's try one of the availabe tools to be used for such problems:

3*2^x + 1 = y^2
3*2^x = y^2 - 1
3*2^x = (y-1)(y+1)

Discussable cases:

a). (y-1) = 3*2^a, (y+1) = 2^b
3*2^(a-1) = 2^(b-1)-1
b). (y+1) = 3*2^a, (y-1) = 2^b
3*2^(a-1) = 2^(b-1)+1
where a + b = x,

Solutions: (0, -2); (0, 2); (3, -5); (3, 5); (4, -7); (4, 7)

[Guest]

The second solution:

Greatest common divisor(y-1,y+1) = 2 that automatically leads to a small number of variants as
you have seen in the previous variant.