2020 IB Extended Essays

3.0 Pell’s Equation

Pell’s equation is the equation 2 − 2 = 1 , where is a square-free positive integer and , are integers (Brilliant.org., [unknown]).

3.1 Fundamental Observations The trivial solution = ±1, = 0 is set aside and the focus is instead on the non-trivial solutions. Furthermore, is conditioned to be a square-free integer (an integer not divisible by a square integer larger than 1) because, otherwise, would render the LHS (left-hand-side) reducible—a polynomial that can be factored into nontrivial polynomials over the ring of, in this case, integers (Weisstein, Eric W [unknown]); if is a square integer such that = 2 ( ∈ ℤ ) , the equation 2 − 2 = 2 − 2 2 = 1 could be reduced to ( + )( − ) = 1 . Therefore, the solutions to equations in the form of Pell’s equation with square integer ’s would be given as = 1, = 0 and = − 1, = 0 . As a result, for all positive integer ’s that are not square-free, Pell’s equation has no solutions other than the trivial. Therefore, assume is square-free. must conditioned to be a positive integer, too, because = 0 would turn the equation into the simple parabolic 2 = 1 , to which the solutions are simply = ±1 , and ∀ ⋖ 0 , the Pell’s equation has no other solutions than the trivial.

6