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i want to find solutions to the equations:

1. [tex]\left(3^x - 2^y\right)^2 = 1[/tex] (x and y are integers)

2. [tex]a^3 - 1 = b^2[/tex] (solutions should be positive integers)

i can "see" that two solutions of the first equation are (1, 1) and (2, 3)

but how can i find the other solutions?

i have seen an equation similar to the second one: [itex]a^3 - 2 = b^2[/itex] which have only 2 solutions (3, 5) and (3, -5). but how do i solve the equation where the constant is -1? is there a general way to solve equations of the form [itex]a^3 + m = b^2[/itex]?

as i am self studying, these equations seem really complicated for me to solve, although experts on number theory might find them trivial. i apologize to them for asking such trivial (to them) questions.

1. [tex]\left(3^x - 2^y\right)^2 = 1[/tex] (x and y are integers)

2. [tex]a^3 - 1 = b^2[/tex] (solutions should be positive integers)

i can "see" that two solutions of the first equation are (1, 1) and (2, 3)

but how can i find the other solutions?

i have seen an equation similar to the second one: [itex]a^3 - 2 = b^2[/itex] which have only 2 solutions (3, 5) and (3, -5). but how do i solve the equation where the constant is -1? is there a general way to solve equations of the form [itex]a^3 + m = b^2[/itex]?

as i am self studying, these equations seem really complicated for me to solve, although experts on number theory might find them trivial. i apologize to them for asking such trivial (to them) questions.

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